Tangent-point energies and ropelength as Gamma-limits of discrete tangent-point energies on biarc curves

Abstract

Using interpolation with biarc curves we prove Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies in the $C^{1}$-topology, as well as to the ropelength functional. As a consequence, discrete almost minimizing biarc curves converge to minimizers of the continuous tangent-point energies, and to ropelength minimizers, respectively. In addition, taking point-tangent data from a given $C^{1,1}$-curve γ, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of γ, together with an explicit convergence rate.

Introduction

The ropelength¹ of a closed arclength parametrized curve $\gamma :\mathbb{R}/L\mathbb{Z}\to \mathbb{R}^{3}$ is defined as the quotient of its length and thickness,

$$\begin{aligned} \mathcal {R}(\gamma ):=\frac{\mathscr{L}(\gamma )}{\triangle [\gamma ]}= \frac{L}{\triangle [\gamma ]}. \end{aligned}$$

(1)

Here, for variational considerations, the thickness $\triangle [\gamma ]$ is most conveniently expressed following Gonzalez and Maddocks [15]—without any regularity assumptions on the curve γ—as

$$\begin{aligned} \triangle [\gamma ]:=\inf_{s\neq t\neq\tau \neq s}R \bigl(\gamma (s), \gamma (t),\gamma (\tau ) \bigr), \end{aligned}$$

(2)

where $R(x,y,z)$ denotes the circumcircle radius of the three points $x,y,z\in \mathbb{R}^{3}$. Motivated by numerous applications in the natural sciences, ropelength is used in numerical computations (see [2, 10, 11, 20] and the references therein) to mathematically model long and slender objects such as strings or macromolecules that do not self-intersect. In fact, it was proved rigorously in [9, 16] that a curve of finite ropelength is embedded and of class $C^{1,1}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$, which means that its curvature exists and is bounded a.e. on $\mathbb{R}/L\mathbb{Z}$. Moreover, a curve γ with positive thickness $\triangle [\gamma ]>0$ is surrounded by an embedded tube with radius equal to $\triangle [\gamma ]$ as shown in [16, Lemma 3], which justifies the use of the non-smooth quantity $\triangle [\cdot ]$ as a steric excluded volume constraint.

The minimization over all triples of curve points to evaluate thickness in (2) is costly, which leads to the idea to replace minimization by integration; see [15, p. 4773]. One such integral energy is the tangent-point energy

$$\begin{aligned} \mathrm {TP}_{q}(\gamma ):= \iint _{(\mathbb{R}/L\mathbb{Z})^{2}} \frac{1}{r^{q}_{\mathrm{ {tp}}}(\gamma (s),\gamma (t))}\,\mathrm {d}s \,\mathrm {d}t,\quad q\ge 2, \end{aligned}$$

(3)

where the circumcircle radius is now replaced by the tangent-point radius

$$\begin{aligned} r_{\mathrm{ {tp}}} \bigl(\gamma (s),\gamma (t) \bigr)= \frac{ \vert \gamma (s)-\gamma (t) \vert ^{2}}{2\operatorname{dist}(\gamma (s)+\mathbb{R}\gamma '(s),\gamma (t))}, \end{aligned}$$

(4)

i.e., the radius of the unique circle through the points $\gamma (s)$ and $\gamma (t)$ that is in addition tangent to the curve γ at $\gamma (s)$. Also, this energy implies self-avoidance and has regularizing properties. It was shown in [33] that if $\mathrm {TP}_{q}(\gamma )$ is finite for some $q>2$, then γ is embedded and of class $C^{1,1-\frac{2}{q}}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$. Later, Blatt [5] improved this regularity to the optimal fractional Sobolev² regularity $W^{2-\frac{1}{q},q}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$, which actually characterizes curves of finite $\mathrm {TP}_{q}$-energy. The knowledge of the exact energy space was then used to establish continuous differentiability of the tangent-point energy [7, Remark 3.1], [34], and to find $\mathrm {TP}_{q}$-critical knots by means of Palais’s symmetric criticality principle [14]. Very recently, long-time existence for a suitably regularized gradient flow for $\mathrm {TP}_{q}$ was shown via a minimizing movement scheme [22].

But the tangent-point energy was also used in numerical simulations. Bartels et al. added a desingularized variant of the $\mathrm {TP}_{q}$-energy in [3, 4] as a self-avoidance term to the bending energy to find elastic knots. The impressive simulations of Crane et al. in [36] use the $\mathrm {TP}_{q}$-energy as well to avoid self-intersections, a higher-dimensional tangent-point energy allows for computations on self-avoiding surfaces; see [35].

In the present paper we address the mathematical question of variational convergence of suitably discretized tangent-point energies towards the continuous $\mathrm {TP}_{q}$-energy, as well as towards ropelength. To account for the tangential information encoded in the tangent-point radius in (3) on the discrete level we use biarcs, i.e., pairs of circular arcs, which on the one hand, can interpolate point-tangent data

$$\begin{aligned} \bigl( \gamma (s_{i}),\gamma '(s_{i}) \bigr) \in \mathbb{R}^{3}\times {\mathbb{S}}^{2} \quad \text{for }i=1,\ldots,n \end{aligned}$$

(5)

of a given arclength parametrized curve $\gamma \in C^{1}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$. Every biarc curve β consisting of n consecutive biarcs is therefore a $C^{1,1}$-interpolant of the curve γ. On the other hand, every biarc curve produces point-tangent data

$$\begin{aligned} (q_{i},t_{i})\in \mathbb{R}^{3} \times {\mathbb{S}}^{2} \quad \text{for }i=1, \ldots,n, \end{aligned}$$

(6)

on its own, namely the points $q_{i}$ and unit-tangents $t_{i}$ at every junction of two consecutive biarcs. In principle, we believe that one could carry out the analysis with other splines, but biarcs are well adapted to discretizing geometric curvature energies such as the tangent-point energy or ropelength. The respective integrands are defined by circles, and relevant geometric quantities like arclength, curvature, torsion, or the global radius of curvature can be evaluated accurately pointwise everywhere on biarc curves, thus providing rigorous upper bounds for the ropelength of ideal knots, see [11, p. 10], [31, p. 81], [17]. In order to avoid degeneracies we restrict ourselves to those biarc curves β whose biarcs have lengths $\lambda _{i}$ that are controlled in terms of the curve’s length $\mathscr{L}(\gamma )$ by means of the inequality

$$\begin{aligned} \frac{\mathscr{L}(\gamma )}{2n}\le \lambda _{i}\le \frac{2\mathscr{L}(\gamma )}{n}\quad \text{for }i=0,\ldots,n-1. \end{aligned}$$

(7)

Let $\mathcal {B}_{n}$ be the class of biarc curves β satisfying (7). Accordingly, we define in a parameter-invariant fashion the discrete tangent-point energy $\mathcal {E}_{q}^{n}$ for $n\in \mathbb{N}$ and $q\in [2,\infty )$ on closed $C^{1}$-curves γ as

$$\begin{aligned} \mathcal {E}_{q}^{n}(\gamma ):=\textstyle\begin{cases} \sum_{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} \left(\frac{2 \operatorname{dist}(l(q_{j}),q_{i})}{ \vert q_{i}-q_{j} \vert ^{2}} \right)^{q} \lambda _{i} \lambda _{j} & \text{if }\gamma \in \mathcal{B}_{n},\\ \infty & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

(8)

with the straight lines $l(q_{i}):=q_{i} + \mathbb{R}t_{i}$ for $i=0,\ldots, n-1$. Note that both $\mathrm {TP}_{q}$ and $\mathcal {E}_{q}^{n}$ are invariant under reparametrization of the curves, and they have the same scaling behavior,

$$\begin{aligned} \mathrm {TP}_{q}(d\gamma )=d^{2-q}\mathrm {TP}_{q}(\gamma )\quad \text{and}\quad \mathcal {E}_{q}^{n}(d \gamma )=d^{2-q}\mathcal {E}_{q}^{n}(\gamma ) \quad \text{for any }d>0. \end{aligned}$$

(9)

We restrict ourselves to injective $C^{1}$-curves that are parametrized by arclength, denoted as the subset $C^{1}_{\mathrm{ {ia}}}$ to state our main results.

Theorem 1.1

(Γ-convergence to tangent-point energy)

For $q>2$ and $L>0$ the discrete tangent-point energies $\mathcal {E}^{n}_{q}$ Γ-converge to the tangent-point energy $\mathrm {TP}_{q}$ on the space $C^{1}_{\mathrm{ {ia}}}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$ with respect to the $\|\cdot \|_{C^{1}}$-norm as $n\to \infty$, i.e.,

$$\begin{aligned} \mathcal {E}_{q}^{n} \stackrel{\Gamma}{ \underset{n \to \infty}{\longrightarrow}} \mathrm {TP}_{q}\quad \textit{on } \bigl(C_{\mathrm{ {ia}}}^{1} \bigl(\mathbb{R}/L\mathbb{Z}, \mathbb{R}^{3} \bigr), \Vert \cdot \Vert _{C^{1}} \bigr). \end{aligned}$$

(10)

As an immediate consequence we infer the convergence of almost minimizers in a given knot class $\mathcal {K}$ of the discrete energies $\mathcal {E}^{n}_{q}$ to a minimizer of the continuous tangent-point energy $\mathrm {TP}_{q}$ in the same knot class $\mathcal {K}$.

Corollary 1.2

(Convergence of discrete almost minimizers)

Let $q>2$, $L>0$, and $\mathcal{K}$ be a tame knot class and $b_{n} \in \mathcal{C}^{*}:=C_{\mathrm{ {ia}}}^{1} (\mathbb{R}/L \mathbb{Z}, \mathbb{R}^{3} )\cap \mathcal{K}$ with

$$\begin{aligned} \Bigl\vert \inf_{\mathcal{C}^{*}} \mathcal {E}_{q}^{n} - \mathcal {E}_{q}^{n} (b_{n} ) \Bigr\vert \to 0\quad \textit{and} \quad \Vert b_{n} - \gamma \Vert _{C^{1}} \to 0\quad \textit{as } n \to \infty. \end{aligned}$$

Then, γ is a minimizer of $\mathrm {TP}_{q}$ in $\mathcal{C}^{*}$ and $\lim_{n \to \infty} \mathcal {E}_{q}^{n}(b_{n})= \mathrm {TP}_{q}(\gamma )$. Furthermore, it holds that $\gamma \in W^{2-\frac{1}{q},q} (\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$.

Moreover, the discrete tangent-point energies can also be used to approximate the non-smooth ropelength functional $\mathcal {R}$ in the sense of Γ-convergence.

Theorem 1.3

(Γ-convergence to ropelength)

It holds that $L^{\frac{n-2}{n}} (\mathcal {E}_{n}^{n} )^{\frac{1}{n}} \stackrel{\Gamma}{\underset{n \to \infty}{\longrightarrow}} \mathcal {R}$ on $(C_{\mathrm{ {ia}}}^{1,1}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3}),\|\cdot \|_{C^{1}} )$.

Also, here we can state the convergence of almost minimizers to ropelength-minimizing curves in a prescribed knot class, which could be of computational relevance for the minimization of ropelength.

Corollary 1.4

(Discrete almost minimizers approximate ropelength minimizers)

Let $\mathcal{K}$ be a tame knot class and $b_{n}, \gamma \in \mathcal{C}^{**}:= C_{\mathrm{ {ia}}}^{1,1} ( \mathbb{R}/ L\mathbb{Z}, \mathbb{R}^{3} )\cap \mathcal{K}$ with

$$\begin{aligned} \Bigl\vert \inf_{ \mathcal{C}^{**}} \mathcal {E}_{n}^{n} - \mathcal {E}_{n}^{n} (b_{n} ) \Bigr\vert \to 0\quad \textit{and} \quad \Vert b_{n} - \gamma \Vert _{C^{1}} \to 0 \quad\textit{as } n \to \infty. \end{aligned}$$

Then, γ is a minimizer of $\mathcal {R}$ in $\mathcal{C}^{**}$ and $\lim_{n \to \infty} L^{\frac{n-2}{n}} (\mathcal {E}_{n}^{n})^{ \frac{1}{n}}(b_{n})=\mathcal {R}(\gamma )$.

To the best of our knowledge, the only known contributions on variational convergence of discrete energies to continuous knot energies are the Γ-convergence results of Scholtes and Blatt. In [26] Scholtes proved the Γ-convergence of a discrete polygonal variant of the Möbius energy to the classic Möbius energy introduced by O’Hara [23]. This result was strengthened later by Blatt [6]. In [27, 28] Scholtes proved the Γ-convergence of polygonal versions of ropelength and of integral Menger curvature to ropelength and to continuous integral Menger curvature, respectively. It remains open at this point if stronger types of variational convergence such as Hausdorff convergence of sets of almost minimizers can be shown for the nonlocal knot energies treated here, as was, e.g., established in [29] for the classic bending energy under clamped boundary conditions. It would be also interesting to set up a numerical scheme for the discretized tangent-point energies $\mathcal {E}^{n}_{q}$ to numerically approximate ropelength minimizers, in comparison to the simulated annealing computations in [11, 31], or to compute discrete (almost) minimizers of the tangent-point energy. The almost linear energy convergence rate established in Theorem 3.1 in Sect. 3 is identical with the one in [26, Proposition 3.1] for Scholtes’ polygonal Möbius energy, which exceeds the $n^{-\frac{1}{4}}$-convergence rate for the minimal distance approximation of the Möbius energy by Rawdon and Simon [24, Theorem 1].

The present paper is structured as follows. In Sect. 2 we provide the necessary background on biarcs—mainly following Smutny’s work [31]. Section 3 is devoted to the convergence of the discretized energies $\mathcal {E}^{n}_{q}$ including explicit convergence rates; see Theorem 3.1. In Sect. 4 we treat Γ-convergence towards the continuous tangent-point energies, as well as convergence of discrete almost minimizers, to prove Theorem 1.1 and Corollary 1.2. Finally, in Sect. 5 we prove Γ-convergence to the ropelength functional, Theorem 1.3 and convergence of discrete almost minimizers to ropelength minimizers, Corollary 1.4. In Appendix A we establish the convergence of rescaled and reparametrized convolutions in fractional Sobolev spaces. Appendix B contains some quantitative analysis of general $C^{1}$-curves.

Biarcs and Biarc curves

The discrete tangent-point energy defined in (8) of the introduction is defined on biarc curves, which are space curves assembled from biarcs, i.e., from pairs of circular arcs. In this section we first present the basic definitions and a general existence result due to Smutny [31, Chap. 4], before specializing to the balanced proper biarc interpolations needed in our convergence proofs later.

Definition 2.1

(Point-tangent pairs and biarcs)

Let $\mathcal{T}:=\mathbb{R}^{3} \times {\mathbb{S}}^{2}$ be the set of point-tangent data $[q,t]$, where ${\mathbb{S}}^{2}$ is the unit sphere in $\mathbb{R}^{3}$.

(i) A point-tangent pair is a pair of tuples of the form $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ with $q_{0} \neq q_{1}$.

(ii) A biarc $(a,\bar{a})$ is a pair of circular arcs in $\mathbb{R}^{3}$ that are continuously joined with continuous tangents and that interpolate a point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$. The common end point m of the two circular arcs a and ā is called the matching point. The interpolation is meant with orientation, such that $t_{0}$ points to the interior of the arc a and $-t_{1}$ points to the interior of the arc ā; see Fig. 1.

Figure 1

Left and middle: Examples of biarcs, $t_{m}$ is the tangent at the common matching point m. Right: The circles $C_{0}$, $C_{1}$, $C_{+}$, and $C_{-}$ of Definition 2.2. Images taken from [31, Figs. 4.1 and 4.3] by courtesy of Jana Smutny

For two points $q_{0}, q_{1} \in \mathbb{R}^{3}$ we set $d:=q_{1} - q_{0}$ and $e:=\frac{q_{1}-q_{0}}{|q_{1}-q_{0}|}=\frac{d}{|d|}$, and define

$$\begin{aligned} R(e):= 2 e \otimes e - {\mathrm {Id}}=2e e^{T}-{\mathrm {Id}}, \end{aligned}$$

(11)

which is a symmetric, proper rotation matrix representing the reflection at the unit vector e. Moreover, for a point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ we set

$$\begin{aligned} t_{0}^{*}:=R(e)t_{0}\quad \text{and}\quad t_{1}^{*}:=R(e)t_{1}. \end{aligned}$$

(12)

Definition 2.2

Let $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ be a point-tangent pair.

(i) Let $\mathcal{C}_{0}$ be the circle through $q_{0}$ and $q_{1}$ with tangent $t_{0}$ at $q_{0}$ and let $\mathcal{C}_{1}$ be the circle through both points with tangent $t_{1}$ at $q_{1}$. If $t_{0} + t_{1}^{*} \neq 0$, we denote the circle through $q_{0}$ and $q_{1}$ with tangent $t_{0} + t_{1}^{*}$ at $q_{0}$ by $\mathcal{C}_{+}$, if $t_{0} - t_{1}^{*} \neq 0$, we denote the circle through both points with tangent $t_{0} - t_{1}^{*}$ at $q_{0}$ by $\mathcal{C}_{-}$; see Fig. 1 on the right.

(ii) A point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ is called cocircular, if $\mathcal{C}_{0}=\mathcal{C}_{1}$ as point sets. A cocircular point-tangent pair is classified as compatible, if the orientations of the two circles induced by the tangents agree, and incompatible otherwise.

Remark 2.3

For a point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$, the compatible cocircular case is equivalent to $t_{0} - t_{1}^{*} =0$. In this case, the circle $\mathcal{C}_{-}$ is not defined. The incompatible cocircular case is equivalent to $t_{0} + t_{1}^{*}=0$, thus the circle $\mathcal{C}_{+}$ is not defined.

The following central existence result of Smutny not only states that interpolating biarcs always exist, but it also characterizes geometrically the possible locations of the corresponding matching points depending on the type of the point-tangent pair. For the precise statement we denote for an arbitrary circle $\mathcal{C}$ through $q_{0}$ and $q_{1}$ the punctured set $\mathcal{C}':=\mathcal{C} \setminus \{q_{0},q_{1}\}$.

Proposition 2.4

([31, Proposition 4.7])

For a given point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$, we denote by $\Sigma _{+} \subset \mathbb{R}^{3}$ the set of matching points of all possible biarcs interpolating the point-tangent pair. Then:

• (i) If $( [q_{0},t_{0} ], [q_{1},t_{1} ] )$ is not cocircular, then $\Sigma _{+}= \mathcal{C}_{+}^{\prime }$.

• (ii) If $( [q_{0},t_{0} ], [q_{1},t_{1} ] )$ is cocircular, we distinguish between two cases:

  • (a) If the point-tangent pair is compatible, then $\Sigma _{+}= \mathcal{C}_{+}^{\prime }=\mathcal{C}_{0}^{\prime }=\mathcal{C}_{1}^{\prime }$.

  • (b) If the point-tangent pair is incompatible, then $\Sigma _{+}$ is the sphere passing through $q_{0}$ and $q_{1}$ perpendicular to the circle $\mathcal{C}_{-}$ without the points $q_{0}$ and $q_{1}$.

• (iii) $\Sigma _{+}$ is a straight line passing through $q_{0}$ and $q_{1}$ without the two points if and only if $t_{0}=t_{1}$ and $\langle t_{0}, e \rangle \neq 0$.

• (iv) $\Sigma _{+}$ is a plane through $q_{0}$ and $q_{1}$ without the two points if and only if $t_{0}=t_{1}$ and $\langle t_{0}, e \rangle =0$.

A particularly powerful interpolation is possible if the location of the matching point $m\in \Sigma _{+}$ of the biarc is roughly “in between” the points $q_{0}$ and $q_{1}$. The following definition states this precisely for the relevant cases (i), (ii)(a), and (iii) of Proposition 2.4.

Definition 2.5

(Desired matching point location $\Sigma _{++}$ and proper biarcs)

(i) Let $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ be a point-tangent pair that is not incompatible cocircular. Then, we denote by $\Sigma _{++} \subset \Sigma _{+}$ the subarc of $\Sigma _{+}$ from $q_{0}$ to $q_{1}$ with the orientation induced by the tangent $t_{0} + t_{1}^{*}$ (see Fig. 2).

Figure 2

The set $\Sigma _{++}$ (boldface) contained in $\Sigma _{+}$: Left: not cocircular and $t_{0}\neq t_{1}$. Right: $t_{0} = t_{1}$ and $\langle t_{0}, e \rangle \neq 0$

(ii) A point-tangent pair $( [q_{0},t_{0} ], [q_{1},t_{1} ] ) \in \mathcal{T} \times \mathcal{T}$ is called proper if $\langle q_{1} - q_{0}, t_{0} \rangle >0$ and $\langle q_{1} - q_{0}, t_{1} \rangle >0$.

(iii) A biarc is called proper if it interpolates a proper point-tangent pair with a matching point $m \in \Sigma _{++}$.

(iv) Let $\gamma \in C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$. We call a biarc γ-interpolating and balanced if, for given $h>0$ and $s \in \mathbb{R}$, it interpolates a point-tangent pair $( [\gamma (s),\gamma '(s) ], [\gamma (s+h), \gamma '(s+h) ] )$, such that the matching point $m_{h} \in \Sigma _{++}^{h}$ satisfies $|m_{h}-\gamma (s)|=|\gamma (s+h)-m_{h}|$, where we indicate the dependence of matching point and location by the index h.

Item (iv) of Definition 2.5 requires that the matching point $m_{h}$ bisects the segment connecting $\gamma (s)$ and $\gamma (s+h)$. That this is indeed possible for sufficiently small h is the content of the following result. Note that here and throughout the paper we use the periodic norm

$$\begin{aligned} \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}:=\min_{k \in \mathbb{Z}} \vert s+Lk-t \vert \end{aligned}$$

(13)

to measure distances in the periodic domain $\mathbb{R}/L\mathbb{Z}$. Moreover, for a continuous function f on $[0,L]$ we denote by $\omega _{f}:[0,L]\to [0,\infty )$ its modulus of continuity, which satisfies $\omega _{f}(0)=0$ and that can be chosen to be concave and non-decreasing.

Lemma 2.6

(Existence of γ-interpolating proper biarc)

Let $\gamma \in C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ and $h\in (0,\frac{L}{2}]$ such that $\omega _{\gamma '}(h)<\frac{1}{2}$. Then, there exists a proper γ-interpolating balanced biarc interpolating the point-tangent pair $( [\gamma (s),\gamma '(s) ], [\gamma (s+h), \gamma '(s+h) ] )$ for all $s\in \mathbb{R}$.

Proof

First, note that (79) and (80) of Lemma B.2 together with $\omega _{\gamma '}(h)<\frac{1}{2}$ and the injectivity of γ imply that

$$\begin{aligned} \bigl\langle \gamma '(s), \gamma (s+h) - \gamma (s) \bigr\rangle >0\quad \text{and}\quad \bigl\langle \gamma '(s+h), \gamma (s+h) - \gamma (s) \bigr\rangle >0. \end{aligned}$$

(14)

Thus, the point-tangent pair is proper according to Definition 2.5(ii).

If the point-tangent pair is not incompatible cocircular and $\gamma '(s)\neq \gamma '(s+h)$ holds, it follows from Proposition 2.4(i) and (ii) (a) that $\Sigma _{+}^{h}$ is the circle $\mathcal{C}_{+}^{\prime }$. Hence, $\Sigma _{++}^{h}$ is a circular arc between $\gamma (s)$ and $\gamma (s+h)$. Thus, the matching point $m_{h}$ can be chosen in $\Sigma _{++}^{h}$ such that $\vert m_{h} - \gamma (s) \vert = \vert \gamma (s+h)- m_{h} \vert$ holds.

If $\gamma '(s)=\gamma '(s+h)$ holds, $\Sigma _{+}^{h}$ is a straight line as a consequence of Proposition 2.4(iii), since we obtain $\langle \gamma '(s), e_{h} \rangle >0$ by dividing (14) through $|\gamma (s+h)-\gamma (s)|$, thus excluding case (iv) of Proposition 2.4. Moreover, with $\gamma '(s)=\gamma '(s+h)$ we infer for the unit vector $e_{h}:=\gamma (s+h)-\gamma (s)/|\gamma (s+h)-\gamma (s)|$ by means of (11) and (12)

$$\begin{aligned} \gamma '(s)+ \bigl(\gamma '(s+h) \bigr)^{*} &\stackrel {\text{(11)},\text{(12)}}{=} \gamma '(s) + 2 (e_{h} \otimes e_{h} ) \gamma '(s+h) - \gamma '(s+h) \\ &\stackrel {}{=} 2 \bigl\langle \gamma '(s), e_{h} \bigr\rangle e_{h}. \end{aligned}$$

Thus, the vector $\gamma '(s)+ (\gamma '(s+h) )^{*}$ is a positive multiple of the vector $e_{h}$. In particular, $\gamma '(s)+ (\gamma '(s+h) )^{*}$ has the same orientation as $e_{h}$. According to Definition 2.5, $\Sigma _{++}^{h}$ is in this case the line segment between $\gamma (s)$ and $\gamma (s+h)$. Therefore, the matching point $m_{h}$ in $\Sigma _{++}^{h}$ can be also chosen such that $\vert m_{h}-\gamma (s) \vert = \vert \gamma (s+h)-m_{h} \vert$.

Hence, we have completed the proof once we have shown that the smallness condition on h excludes case (ii) (b) of Proposition 2.4. Indeed, suppose that the point-tangent pair was incompatible cocircular. Then,

$$\begin{aligned} \gamma '(s) + \bigl(\gamma '(s+h) \bigr)^{*}=0, \end{aligned}$$

(15)

and using (12) we can write

$$\begin{aligned} \bigl(\gamma '(s+h) \bigr)^{*} = 2 e_{h} \otimes e_{h} \gamma '(s+h) - \gamma '(s+h) = 2 \bigl\langle e_{h}, \gamma '(s+h) \bigr\rangle e_{h} - \gamma '(s+h). \end{aligned}$$

This representation inserted into (15) leads to $\gamma '(s+h)-\gamma '(s) = 2 \langle e_{h}, \gamma '(s+h) \rangle e_{h}$ and hence,

$$\begin{aligned} \bigl\vert \gamma '(s+h) - \gamma '(s) \bigr\vert =2 \bigl\vert \bigl\langle e_{h}, \gamma '(s+h) \bigr\rangle \bigr\vert \underbrace{ \vert e_{h} \vert }_{=1}= 2 \frac{ \langle \gamma (s+h) - \gamma (s),\gamma '(s+h) \rangle}{ \vert \gamma (s+h) - \gamma (s) \vert }. \end{aligned}$$

(16)

By virtue of inequality (80) in Lemma B.2 we conclude that

$$\begin{aligned} 2 \bigl(1-\omega _{\gamma '}(h) \bigr) \stackrel {\text{(80)}}{\leq } 2 \frac{ \langle \gamma (s+h) - \gamma (s),\gamma '(s+h) \rangle}{ \vert \gamma (s+h) - \gamma (s) \vert } \stackrel {\text{(16)}}{=} \bigl\vert \gamma '(s+h) - \gamma '(s) \bigr\vert \leq \omega _{\gamma '}(h), \end{aligned}$$

which is equivalent to $\omega _{\gamma '}(h)\geq \frac{2}{3}$, contradicting our assumption on h. □

Glueing together finitely many interpolating biarcs in a $C^{1}$-fashion produces biarc curves precisely defined as follows.

Definition 2.7

([31, cf. Definition 6.1])

(i) A closed biarc curve $\beta: J \to \mathbb{R}^{3}$ is a closed curve assembled from biarcs in a $C^{1}$-fashion where the biarcs interpolate a sequence $( [q_{i},t_{i} ] )_{i \in I}$ of point-tangent tuples. J is a compact interval, $I \subset \mathbb{N}$ bounded, and the first and last point-tangent tuple coincide. The set of such biarc curves is denoted by $\tilde{\mathcal {B}}_{n}$ where n is the number of indices contained in I.

(ii) We call a closed biarc curve proper if every biarc of the curve is proper.

(iii) A biarc curve is γ-interpolating and balanced for a given curve $\gamma \in C^{1}_{\mathrm{ {ia}}}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$ if every biarc of the curve is γ-interpolating and balanced.

Note that the set $\mathcal {B}_{n}$ of closed biarc curves satisfying (7) introduced in the introduction is a strict subset of $\tilde{\mathcal {B}}_{n}$.

Under suitable control of partitions of the periodic domain we can prove the existence of proper, γ-interpolating, and balanced biarc curves in Lemma 2.9 below.

Definition 2.8

Let $c_{1},c_{2} >0$. A sequence $(\mathcal{M}_{n} )_{n \in \mathbb{N}}$ of partitions of $\mathbb{R}/ L\mathbb{Z}$ with $\mathcal{M}_{n}:= \{s_{n,0}, \dots, s_{n,n} \}$ and $0=s_{n,0} < s_{n,1} < \cdots < s_{n,n-1}<s_{n,n}=L$ is called $(c_{1}-c_{2})$-distributed if for

$$\begin{aligned} h_{n}:=\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \quad \text{and}\quad \tilde{h}_{n}:=\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \end{aligned}$$

one has

$$\begin{aligned} \frac{c_{1}}{n}\leq \tilde{h}_{n} \leq h_{n} \leq \frac{c_{2}}{n} \quad\text{and}\quad h_{n} \leq \frac{L}{2} \quad \text{for any }n\in \mathbb{N}. \end{aligned}$$

(17)

Lemma 2.9

Let $\gamma \in C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ $c_{1},c_{2} >0$ and $(\mathcal{M}_{n} )_{n \in \mathbb{N}}$ a sequence of $(c_{1}-c_{2})$-distributed partitions. Then, there is some $N\in \mathbb{N}$ such that for all $n\ge N$ there exists a proper γ-interpolating and balanced biarc curve $\beta _{n}$ interpolating the point-tangent pairs

$$\begin{aligned} \bigl( \bigl( \bigl[\gamma (s_{n,i} ),\gamma ' ( s_{n,i} ) \bigr], \bigl[\gamma (s_{n,i+1} ), \gamma ' (s_{n,i+1} ) \bigr] \bigr) \bigr)_{i=0, \dots, n-1}. \end{aligned}$$

Proof

By means of the defining inequality (17) for the $(c_{1}-c_{2})$-distributed sequence $(\mathcal{M}_{n} )_{n \in \mathbb{N}}$ we have $|s_{n,i+1}-s_{n,i}|_{ \mathbb{R}/ L\mathbb{Z}}=|s_{n,i+1}-s_{n,i}|$ for all $n\in \mathbb{N}$ and $i=0, \dots, n-1$ (see (13)), and we can choose $N\in \mathbb{N}$ so large that the inequalities $\omega _{\gamma '}( \frac{c_{2}}{N} ) < \frac{1}{2}$ and $\frac{c_{2}}{N} \leq \frac{L}{2}$ hold. Then, in particular,

$$\begin{aligned} \omega _{\gamma '}\bigl( \vert s_{n,i+1}-s_{n,i} \vert \bigr) \leq \omega _{\gamma '}(h_{n})\stackrel {\text{(17)}}{\leq } \omega _{\gamma '}\biggl( \frac{c_{2}}{N} \biggr)< \frac{1}{2}\quad \text{for any }n\ge N. \end{aligned}$$

As a consequence of Lemma 2.6, there exists for all $n\ge N$ and $i=0, \dots, n-1$ a proper γ-interpolating and balanced biarc interpolating the point-tangent pair $( [\gamma (s_{n,i} ),\gamma ' ( s_{n,i} ) ], [\gamma (s_{n,i+1} ), \gamma ' (s_{n,i+1} ) ] )$. Now, we assemble for $i=0, \dots, n-1$ these n biarcs as in Definition 2.7 and obtain a biarc curve with the required properties. □

From now on, whenever we write $\beta _{n}$ for a given curve $\gamma \in C^{1}_{\mathrm{ {ia}}}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$, we mean a proper γ-interpolating and balanced biarc curve obtained in Lemma 2.9. By $\lambda _{n,i}$ we denote the length of the ith biarc of the curve $\beta _{n}$. In general, the elements $\beta _{n}$ do not have the same length as the interpolated curve γ. However, Smutny showed in [31] that under certain assumptions the sequence of the lengths $(\mathscr{L} (\beta _{n} ) )_{n \in \mathbb{N}}$ of $\beta _{n}$ converges towards the length $\mathscr{L}(\gamma )$ of γ. The following lemma is an essential ingredient for that proof.

Lemma 2.10

Let $\gamma \in C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ and $(\beta _{n} )_{n \in \mathbb{N}}$ be a sequence of proper γ-interpolating and balanced biarc curves as in Lemma 2.9. Then,

$$\begin{aligned} \lambda _{n,i}- \vert s_{n,i+1}-s_{n,i} \vert =O \bigl( \vert s_{n,i+1}-s_{n,i} \vert ^{3} \bigr)\quad \textit{for all } i=1,\ldots,n-1, \textit{ as } n \to \infty, \end{aligned}$$

where the constant on the right-hand side only depends on the Lipschitz constant of $\gamma '$.

Proof

We identify the periodic domain $\mathbb{R}/L\mathbb{Z}$ with $[0,L]$ and check that $(c_{1}-c_{2})$-distributed partitions of $\mathbb{R}/L\mathbb{Z}$ satisfy Smutny’s requirements in [31, Notation 6.2, 6.3] apart from the nestedness of the mesh. The latter, however, is not necessary in her proof; whence we can apply [31, Lemma 6.8] to conclude the statement, where the dependence of the constant follows from the proof of [31, Lemma 6.8]. □

Now, we show that the lengths $\mathscr{L}(\beta _{n})$ of proper γ-interpolating and balanced biarc curves $\beta _{n}$ converge towards the length $\mathscr{L}(\gamma )$ of γ.

Theorem 2.11

Let $\gamma \in C^{1,1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ and $(\beta _{n} )_{n \in \mathbb{N}}$ be a sequence of proper γ-interpolating and balanced biarc curves. Then, $| \frac{\mathscr{L} (\beta _{n} )}{\mathscr{L}(\gamma )} -1 | \to 0$ as $n \to \infty$.

Proof

This follows directly from [31, Corollary 6.9]; under the same preconditions as we verified in the proof of Lemma 2.10. □

In order to address convergence of biarc curves $\beta _{n}$ to the interpolated curve γ we need to reparametrize $\beta _{n}$ for all $n \in \mathbb{N}$ such that those reparametrizations are defined on $\mathbb{R}/ L\mathbb{Z}$ like γ is. An explicit reparametrization function that maps the arclength parameters of γ at the supporting points of the mesh to the arclength parameters of $\beta _{n}$ is constructed in [31, Appendix A]. With that, we can show the $C^{1}$-convergence of a reparametrized sequence of biarc curves to the interpolated curve γ.

Theorem 2.12

Let $\gamma \in C^{1,1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$, and let $(\beta _{n} )_{n \in \mathbb{N}}$ be a sequence of proper γ-interpolating and balanced biarc curves parametrized by arclength. Then, for $B_{n}:=\beta _{n} \circ \varphi _{n}$ with $\varphi _{n}$ as constructed in [31, Appendix A] one has $\|\gamma -B_{n}\|_{C^{1}} \to 0$ as $n \to \infty$.

Proof

We want to apply [31, Theorem 6.13], where Smutny showed $C^{1}$-convergence under certain assumptions. Additionally to the hypotheses checked before in the proof of Lemma 2.10, we need to show that the so-called biarc parameters $\Lambda _{n,i}$ of the ith biarc of the biarc curve $\beta _{n}$, representable as (cf. [31, Lemma 4.13])

$$\begin{aligned} \Lambda _{n,i}= \frac{\langle \gamma '(s_{n,i}),\gamma (s_{n,i+1})-\gamma (s_{n,i})\rangle \vert m_{n,i}-\gamma (s_{n,i}) \vert ^{2}}{\langle \gamma '(s_{n,i}), m_{n,i} - \gamma '(s_{n,i}) \rangle \vert \gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert ^{2}}, \end{aligned}$$

where the $m_{n,i}$ are the matching points of the ith biarc, are uniformly bounded from below and from above. In other words, we have to prove that there exist two constants $\Lambda _{\min},\Lambda _{\max}$ such that

$$\begin{aligned} 0 < \Lambda _{\min} \leq \Lambda _{n,i} \leq \Lambda _{\max} < 1 \quad \text{for any }n \in \mathbb{N}, i=0,\dots, n-1. \end{aligned}$$

Using the fact that the biarc curves are balanced, i.e., $|m_{n,i}-\gamma (s_{n,i})|=|m_{n,i}-\gamma (s_{n,i+1})|$, and that γ is parametrized by arclength, we can then estimate by means of (79) in Lemma B.2 in the appendix

$$\begin{aligned} \Lambda _{n,i}&\stackrel {}{\geq }\frac{\langle \gamma '(s_{n,i}),\gamma (s_{n,i+1})-\gamma (s_{n,i})\rangle \vert m_{n,i}-\gamma (s_{n,i}) \vert ^{2}}{ \vert m_{n,i}-\gamma (s_{n,i})\|\gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert ^{2}} \\ &\stackrel {}{=}\frac{\langle \gamma '(s_{n,i}),\gamma (s_{n,i+1})-\gamma (s_{n,i})\rangle}{ \vert \gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert } \frac{ \vert m_{n,i}-\gamma (s_{n,i}) \vert }{ \vert \gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert } \\ &\stackrel {}{\geq }\frac{\langle \gamma '(s_{n,i}),\gamma (s_{n,i+1})-\gamma (s_{n,i})\rangle}{ \vert \gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert } \underbrace{ \frac{ \vert m_{n,i}-\gamma (s_{n,i}) \vert }{ \vert m_{n,i}-\gamma (s_{n,i+1}) \vert + \vert m_{n,i}-\gamma (s_{n,i}) \vert }}_{= \frac{1}{2}} \\ &\overset{\text{(79)}}{\ge} \frac{1}{2} \bigl(1- \omega _{\gamma '}\bigl( \vert s_{n,i+1}-s_{n,i} \vert \bigr) \bigr) \ge \frac{1}{2} \bigl(1-\omega _{\gamma '}(h_{n}) \bigr). \end{aligned}$$

Hence, we can choose n sufficiently large such that $\frac{1}{2}(1-\omega _{\gamma '}(h_{n}))\geq \frac{1}{4}=:\Lambda _{\min}$. On the other hand, by [31, Lemma 5.6] we have

$$\begin{aligned} \Lambda _{n,i}=1-\bar{\Lambda}_{n,i}+O \bigl(h_{n,i}^{2} \bigr)\quad \text{for any }i=0, \ldots,n-1, \text{{ as $n\to \infty $,}} \end{aligned}$$

where the constant hidden in the $O(h_{n,i}^{2})$-term only depends on the curve γ and where $\bar{\Lambda}_{n,i}$ is given by

$$\begin{aligned} \bar{\Lambda}_{n,i} = \frac{\langle \gamma '(s_{n,i+1}),\gamma (s_{n,i+1})-\gamma (s_{n,i})\rangle \vert m_{n,i}- \gamma (s_{n,i+1}) \vert ^{2}}{\langle \gamma '(s_{n,i+1}), \gamma '(s_{n,i+1})-m_{n,i} \rangle \vert \gamma (s_{n,i+1})-\gamma (s_{n,i}) \vert ^{2}}. \end{aligned}$$

As for $\Lambda _{n,i}$, we can estimate $\bar{\Lambda}_{n,i}\geq \frac{1}{2}(1-\omega _{\gamma '}(h_{n}))$, which yields

$$\begin{aligned} \Lambda _{n,i}& \leq 1-\frac{1}{2} \bigl(1-\omega _{\gamma '}(h_{n}) \bigr) + O \bigl(h_{n}^{2} \bigr) \leq \frac{1}{2} \bigl(1+\omega _{\gamma '}(h_{n}) \bigr)+O \bigl(h_{n}^{2} \bigr)\quad \text{{as $n\to \infty $.}} \end{aligned}$$

Hence, again choosing n sufficiently large, we obtain $\frac{1}{2}(1+\omega _{\gamma '}(h_{n}))+O(h_{n}^{2})\leq \frac{3}{4}=: \Lambda _{\max}$ as the necessary uniform bound on the biarc parameters. Therefore, [31, Theorem 6.13] is applicable and we obtain that $B_{n} \to \gamma$ in $C^{1}$ as $n \to \infty$. □

Discrete energies on interpolating biarc curves that converge to the continuous $\mathrm {TP}_{q}$ energy

For the central convergence result of this section, Theorem 3.1, we work with discrete tangent-point energies $\tilde{\mathcal {E}^{n}_{q}}$ with the larger effective domain $\tilde{\mathcal {B}}_{n}$ (see Definition 2.7(i)), instead of with $\mathcal {E}^{n}_{q}$ introduced in (8) of the introduction, whose effective domain $\mathcal {B}_{n}$ is defined by the constraint (7). In other words,

$$\begin{aligned} \tilde{\mathcal {E}}_{q}^{n}(\gamma ):=\textstyle\begin{cases} \sum_{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} \left(\frac{2 \operatorname{dist}(l(q_{j}),q_{i})}{ \vert q_{i}-q_{j} \vert ^{2}} \right)^{q} \lambda _{i} \lambda _{j} & \text{if }\gamma \in \tilde{\mathcal{B}}_{n},\\ \infty & \text{otherwise.} \end{cases}\displaystyle \end{aligned}$$

(18)

These discrete energies evaluated on a sequence $(\beta _{n} )_{n \in \mathbb{N}}$ of proper γ-interpolating and balanced biarc curves converge with a certain rate to the continuous $\mathrm {TP}_{q}$-energy of γ if γ is sufficiently smooth. Some of the ideas in the proof of the theorem are based on [26, Proposition 3.1] by Scholtes.

Theorem 3.1

Let $c_{1},c_{2}>0$ and $\gamma \in C^{1,1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$, and $(\mathcal{M}_{n} )_{n \in \mathbb{N}}$ with $\mathcal{M}_{n}= \{s_{n,0},\dots, s_{n,n} \}$ be a $(c_{1}-c_{2})$-distributed sequence of partitions of $\mathbb{R}/ L\mathbb{Z}$ (see Definition 2.8). Then, there is a constant $C>0$ depending on $q,c_{1},c_{2}$, and γ, such that for a sequence $(\beta _{n} )_{n \in \mathbb{N}}$ of proper γ-interpolating and balanced biarc curves interpolating the point-tangent data

$$\begin{aligned} \bigl( \bigl( \bigl[\gamma (s_{n,i} ),\gamma ' (s_{n,i} ) \bigr], \bigl[\gamma (s_{n,i+1} ),\gamma ' (s_{n,i+1} ) \bigr] \bigr) \bigr)_{i=0, \dots, n-1} \end{aligned}$$

with $\beta _{n} \in \tilde{\mathcal{B}}_{n}$ for all $n \in \mathbb{N}$, there is an index $N \in \mathbb{N}$ with

$$\begin{aligned} \bigl\vert \mathrm {TP}_{q} (\gamma ) -\tilde{\mathcal {E}}_{q}^{n} (\beta _{n} ) \bigr\vert \leq \frac{C \ln (n)}{n} \quad \textit{for any }n\ge N. \end{aligned}$$

Note that in particular, the convergence rate $\frac{\ln (n)}{n}$ implies the convergence rate $\frac{1}{n^{1-\varepsilon}}$ for any given $\varepsilon >0$.

Proof of Theorem 3.1

Set $\Upsilon =4\frac{c_{2}}{c_{1}}$ and define for $i,j \in \{0, \dots, n\}$ the periodic index distance

$$\begin{aligned} \vert i-j \vert _{n}:=\min \bigl\{ \vert i-j \vert ,n- \vert i-j \vert \bigr\} . \end{aligned}$$

We then decompose

$$\begin{aligned} \begin{aligned} \mathrm {TP}_{q} (\gamma )-\tilde{ \mathcal {E}}_{q}^{n} (\beta _{n} ) ={}& \sum _{i=0}^{n-1}\sum_{j, \vert i-j \vert _{n}\leq \Upsilon} \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \biggl(2\frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}} \biggr)^{q} \,\mathrm {d}s \,\mathrm {d}t \\ &{}-\sum_{i=0}^{n-1}\sum _{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \lambda _{n,i}\lambda _{n,j} \\ &{}+2^{q} \sum_{i=0}^{n-1}\sum _{j, \vert i-j \vert _{n}> \Upsilon} (A_{i,j}+B_{i,j}+C_{i,j} ), \end{aligned} \end{aligned}$$

(19)

with

$$\begin{aligned} &A_{i,j}:= \int _{s_{n,j}}^{s_{n,j+1}} \int _{ s_{n,i}}^{s_{n,i+1}} \frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )^{q}-\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s )-\gamma (t ) \vert ^{2q}} \,\mathrm {d}s \, \mathrm {d}t, \\ &B_{i,j}:= \int _{s_{n,j}}^{s_{n,j+1}} \int _{ s_{n,i}}^{s_{n,i+1}} \biggl( \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s )-\gamma (t ) \vert ^{2q}}- \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \biggr) \,\mathrm {d}s \,\mathrm {d}t, \\ &C_{i,j}:= \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert - \lambda _{n,i}\lambda _{n,j} \bigr]. \end{aligned}$$

Step 1: Since γ is an injective $C^{1}$-curve it is bi-Lipschitz (see Lemma B.1), i.e., there exists a constant $c_{\gamma }\in (0,\infty )$ such that

$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}\leq c_{\gamma } \bigl\vert \gamma (t)-\gamma (s) \bigr\vert \quad \text{for any }t, s \in \mathbb{R}. \end{aligned}$$

(20)

Step 2: Now, we give an upper bound for $2 \frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}}$ for all $s,t \in \mathbb{R}$ with $s\neq t$. Without loss of generality we assume $t< s$. Then, there exists a number $k=k(s,t) \in \mathbb{Z}$ satisfying $|t-s|_{ \mathbb{R}/ L\mathbb{Z}}=|kL+t-s|$. We use the periodicity of γ and $K:=\|\gamma ''\|_{L^{\infty}}<\infty$ (since $\gamma \in C^{1,1}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})\simeq W^{1,\infty}( \mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$) to estimate

$$\begin{aligned} & \operatorname{dist}\bigl(l \bigl(\gamma (t ) \bigr),\gamma (s ) \bigr) \\ &\quad=\inf _{\mu \in \mathbb{R}} \bigl\vert \gamma (s)-\gamma (t)-\mu \gamma '(t) \bigr\vert \\ &\quad\le \bigl\vert \gamma (s)-\gamma (kL+t)- \bigl(s-(kL+t) \bigr)\gamma '(kL+t) \bigr\vert \\ &\quad\le \int ^{s}_{kL+t} \int ^{u}_{kL+t} \bigl\vert \gamma ''(v) \bigr\vert \,\mathrm {d}v \,\mathrm {d}u \le K (kL+t-s)^{2}=K \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}, \end{aligned}$$

(21)

where we assumed, without loss of generality, that $kL+t< s$ for the integrals. Therefore, by means of (20)

$$\begin{aligned} \biggl(2\frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}} \biggr)^{q} &\leq \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \quad \text{for any }s,t\in \mathbb{R}, s\neq t. \end{aligned}$$

(22)

Define $C_{1}:= (2 c_{\gamma}^{2} K )^{q} c_{2} (2\Upsilon +1)L$. Applying the calculations above we can estimate the first term on the right-hand side of (19) from above by

$$\begin{aligned} \stackrel {\text{(22)}}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \sum _{i=0}^{n-1}\sum_{j, \vert i-j \vert _{n}\leq \Upsilon} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert \\ \stackrel {\text{(17)}}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \frac{c_{2}}{n} \sum_{i=0}^{n-1} \vert s_{n,i+1}-s_{n,i} \vert \underbrace{ \sum _{j, \vert i-j \vert _{n}\leq \Upsilon} 1}_{\leq 2\Upsilon +1} \\ \stackrel {}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} { \frac{c_{2}}{n}} (2\Upsilon +1) \underbrace{ \sum_{i=0}^{n-1} \vert s_{n,i+1}-s_{n,i} \vert }_{=L} \stackrel {}{=}\frac{C_{1}}{n}. \end{aligned}$$

(23)

Step 3: By Lemma 2.10 there exists a constant $c_{K}$ only depending on K such that

$$\begin{aligned} \biggl\vert \frac{\lambda _{n,i}}{ \vert s_{n,i+1}-s_{n,i} \vert }-1 \biggr\vert \leq c_{K} \vert s_{n,i+1}-s_{n,i} \vert ^{2} \end{aligned}$$

for all $n \geq N$ and $i=0,\dots,n-1$, where N depends on the given sequence of biarc curves. Using the fact that $\vert s_{n,i+1}-s_{n,i} \vert \leq \frac{L}{2}$ yields

$$\begin{aligned} \lambda _{n,i} \leq \underbrace{c_{K} \biggl( \frac{L}{2} \biggr)^{2}}_{=:d_{K}} \vert s_{n,i+1}-s_{n,i} \vert \quad \text{for }n\geq N \text{ and } i=0, \dots, n-1. \end{aligned}$$

(24)

Without loss of generalization we can assume that $d_{K} \geq 1$. Define $C_{2}:=d_{K}^{2} C_{1}$. Thus,

$$\begin{aligned} & \sum_{i=0}^{n-1}\sum _{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \lambda _{n,i}\lambda _{n,j} \\ &\quad \stackrel {\text{(24)}}{\leq } d_{K}^{2} \sum _{i=0}^{n-1}\sum_{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert \\ &\quad \stackrel {\text{(22)}}{\leq } { d_{K}^{2} \bigl(2c_{\gamma}^{2}K \bigr)^{q} \sum _{i=0}^{n-1}\sum_{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert } \overset{\text{(23)}}{\le} \frac{C_{2}}{n}, \end{aligned}$$

(25)

which deals with the second term on the right-hand side of (19).

Step 4: We assume from now on that $|i-j|_{n}>\Upsilon$. The sequence $(\mathcal{M}_{n})_{n}$ is assumed to be $(c_{1}-c_{2})$-distributed, so that in view of (17)

$$\begin{aligned} \vert s_{n,k+1}-s_{n,k} \vert = \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}\quad \text{for any }n\in \mathbb{N}\ \text{and}\ k=0, \dots, n-1. \end{aligned}$$

For $s \in [s_{n,i},s_{n,i+1})$ and $t \in [s_{n,j},s_{n,j+1})$ with $i\neq j$ we use $|s-t|_{\mathbb{R}/L\mathbb{Z}}\le |s-s_{n,i}|_{ \mathbb{R}/ L\mathbb{Z}} + |s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}+|s_{n,j}-t|_{ \mathbb{R}/ L\mathbb{Z}}$ to infer the inequality

$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} & \stackrel {}{\leq }\vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}+2\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \stackrel {\text{(17)}}{\leq } \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} + 2\frac{c_{2}}{c_{1}} \underbrace{ \min _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}}_{\leq \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ & \stackrel {}{\leq }\biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$

From $|i-j|_{n}>\Upsilon =4\frac{c_{2}}{c_{1}}$ we have in particular

$$\begin{aligned} 2\frac{c_{2}}{c_{1}}< \frac{1}{2} \vert i-j \vert _{n}. \end{aligned}$$

(26)

Then, similarly as before,

$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} &\stackrel {}{\geq }\vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} - 2\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\stackrel {\text{(17)}}{\geq } \vert i-j \vert _{n} \min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}-{ \frac{2c_{2}}{c_{1}}}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \stackrel {}{=}\biggl( \vert i-j \vert _{n}-\frac{2c_{2}}{c_{1}} \biggr) \min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \overset{\text{(26)}}{>} \frac{1}{2} \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \stackrel {}{\geq }\frac{c_{1}}{2c_{2}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$

In total, we conclude for $|i-j|_{n}>\Upsilon$

$$\begin{aligned} \frac{c_{1}}{2c_{2}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \leq \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} \leq \biggl(1+2 \frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \end{aligned}$$

(27)

for $s\in [s_{n,i},s_{n,i+1})$ and $t\in [s_{n,j},s_{n,j+1})$, which we consider also in Steps 5 and 6.

Step 5: In order to estimate $A_{i,j}$, we initially estimate for arbitrary $a,b\geq 0$

$$\begin{aligned} \bigl\vert b^{q}-a^{q} \bigr\vert = \biggl\vert \int _{a}^{b} \frac{d}{dx}x^{q} \, \mathrm {d}x \biggr\vert = \biggl\vert \int _{a}^{b} qx^{q-1} \,\mathrm {d}x \biggr\vert \leq q \vert b-a \vert \max \{a,b\}^{q-1}, \end{aligned}$$

(28)

since the function $f: [0,\infty ) \to [0,\infty ), x \to x^{q-1}$ is nondecreasing for $q\geq 2$. We abbreviate $d(\cdot,\cdot ):=\operatorname{dist}(l(\gamma (\cdot )),\gamma (\cdot ))$ and use estimate (28) to find for $s \in [s_{n,i},s_{n,i+1})$ and $t \in [s_{n,j},s_{n,j+1})$

$$\begin{aligned} \bigl\vert d^{q}(t,s)-d^{q}(s_{n,j}, s_{n,i}) \bigr\vert \leq q \bigl\vert d(t,s)-d(s_{n,j},s_{n,i}) \bigr\vert \bigl(\max \bigl\{ d(t,s),d(s_{n,j},s_{n,i}) \bigr\} \bigr)^{q-1}. \end{aligned}$$

(29)

Furthermore, combining (21) with (27) yields

$$\begin{aligned} \begin{aligned} & d(t,s)\overset{\text{ (21)}}{\leq} K \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2} \stackrel {\text{ (27)}}{\leq } K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}, \\ &d(s_{n,j},s_{n,i})\leq K \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}\leq K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}. \end{aligned} \end{aligned}$$

(30)

Hence,

$$\begin{aligned} \bigl(\max \bigl\{ d(t,s),d(s_{n,j},s_{n,i}) \bigr\} \bigr)^{q-1} \leq K^{q-1} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-2}. \end{aligned}$$

(31)

Moreover, we estimate again by virtue of (27) now for $s:=s_{n,i}$

$$\begin{aligned} \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}}\stackrel {\text{(27)}}{\leq } \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}, \end{aligned}$$

and we use (17) to find for $t\in [s_{n,j},s_{n,j+1})$

$$\begin{aligned} \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} &\stackrel {}{\leq }\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\stackrel {\text{(17)}}{\leq }{ \frac{c_{2}}{c_{1}}} \min _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}\stackrel {}{\leq }\biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$

Combining these last two estimates with (27) leads to

$$\begin{aligned} \begin{aligned} &\max \bigl\{ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} + \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}+ \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\quad\leq 3 \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \quad \text{for }s\in [s_{n,i},s_{n,i+1}), t\in [s_{n,j}, s_{n,j+1}). \end{aligned} \end{aligned}$$

(32)

For arbitrary $\tau \in \mathbb{R}$ the mapping $P_{\gamma '(\tau )}: \mathbb{R}^{3}\to \mathbb{R}\gamma '(\tau )$ defined as

$$\begin{aligned} P_{\gamma '(\tau )} (v):= \bigl\langle v, \gamma '(\tau ) \bigr\rangle \gamma '(\tau ),\quad \text{for }v\in \mathbb{R}^{3} \end{aligned}$$

(33)

is the orthogonal projection onto the subspace $\mathbb{R}\gamma '(\tau )$ since $|\gamma '|=1$, and we have

$$\begin{aligned} \bigl\vert P_{\gamma '(\tau )} (v) -v \bigr\vert &\leq \vert w-v \vert \quad \text{for all } w \in \mathbb{R}\gamma '(\tau ), v\in \mathbb{R}^{3}. \end{aligned}$$

(34)

Moreover, we have for any $\tau,\sigma \in \mathbb{R}$

$$\begin{aligned} d(\tau,\sigma )=\operatorname{dist}\bigl(l \bigl(\gamma (\tau ) \bigr),\gamma (\sigma ) \bigr) &= \bigl\vert P_{\gamma '(\tau )} \bigl(\gamma (\sigma )-\gamma (\tau ) \bigr) - \bigl(\gamma (\sigma )-\gamma (\tau ) \bigr) \bigr\vert . \end{aligned}$$

(35)

Furthermore, we calculate for $s\in [s_{n,i},s_{n,i+1})$ and $t\in [s_{n,j},s_{n,j+1})$ using the linearity of the projection

$$\begin{aligned} & P_{\gamma '(t)} \bigl(\gamma (s) -\gamma (t) \bigr)-P_{\gamma ' (s_{n,j} )} \bigl( \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr) \\ &\quad \stackrel {}{=}P_{\gamma '(t)} \bigl(\gamma (s)-\gamma (s_{n,i}) \bigr) +P_{\gamma '(t)} \bigl(\gamma (s_{n,i})-\gamma (t) \bigr) \\ &\qquad{}-P_{\gamma ' (s_{n,j} )} \bigl(\gamma (t)-\gamma (s_{n,j}) \bigr) -P_{\gamma ' (s_{n,j} )} \bigl(\gamma (s_{n,i})-\gamma (t) \bigr) \\ &\quad \stackrel {\text{(33)}}{=} P_{\gamma '(t)} \bigl(\gamma (s)-\gamma (s_{n,i} ) \bigr) - P_{\gamma ' (s_{n,j} )} \bigl( \gamma (t)-\gamma (s_{n,j} ) \bigr) \\ &\qquad{}+ \bigl\langle \gamma (s_{n,i} ) - \gamma (t),\gamma '(t)-\gamma ' (s_{n,j} ) \bigr\rangle \gamma '(t) \\ &\qquad{}+ \bigl\langle \gamma (s_{n,i} ) - \gamma (t), \gamma ' (s_{n,j} ) \bigr\rangle \bigl( \gamma '(t) - \gamma ' (s_{n,j} ) \bigr) . \end{aligned}$$

(36)

In conclusion, by (35) and the elementary inequality $\Vert a|-|b\Vert \le |a-b|$, this yields for the expression $|d(t,s)-d(s_{n,j},s_{n,i})|$ (for $s\in [s_{n,i},s_{n,i+1})$ and $t\in [s_{n,j},s_{n,j+1})$) the upper bound

$$\begin{aligned} \bigl\vert P_{\gamma '(t)} \bigl(\gamma (s) - \gamma (t) \bigr)-P_{ \gamma ' (s_{n,j} )} \bigl(\gamma (s_{n,i} ) - \gamma (s_{n,j} ) \bigr)- \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr)+ \bigl(\gamma (t) - \gamma (s_{n,j} ) \bigr) \bigr\vert , \end{aligned}$$

which in turn by means of (36) and (34) can be bounded from above by

$$\begin{aligned} & \bigl\vert P_{\gamma '(t)} \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr) - \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert P_{\gamma ' (s_{n,j} )} \bigl( \gamma (t) - \gamma (s_{n,j} ) \bigr) - \bigl(\gamma (t) - \gamma (s_{n,j} ) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \bigl\langle \gamma (s_{n,i}) - \gamma (t),\gamma '(t) - \gamma ' (s_{n,j} ) \bigr\rangle \gamma '(t) \bigr\vert \\ &\qquad{}+ \bigl\vert \bigl\langle \gamma (s_{n,i} ) - \gamma (t), \gamma ' (s_{n,j} ) \bigr\rangle \bigl(\gamma '(t) - \gamma ' (s_{n,j} ) \bigr) \bigr\vert \\ &\quad \stackrel {\text{(34)}}{\leq } \bigl\vert \bigl(\gamma (s)- \gamma (s_{n,i} ) \bigr)-(s-s_{n,i})\gamma '(t) \bigr\vert + \bigl\vert \bigl(\gamma (t)- \gamma (s_{n,j} ) \bigr)-(t- s_{n,j})\gamma ' (s_{n,j} ) \bigr\vert \\ &\qquad{}+2 \bigl\vert \gamma (s_{n,i} )-\gamma (t) \bigr\vert \bigl\vert \gamma ' (s_{n,j} )-\gamma '(t) \bigr\vert . \end{aligned}$$

The last summand is bounded by $2K|t-s_{n,i}|_{ \mathbb{R}/ L\mathbb{Z}}|t-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}$ since $K=\|\gamma ''\|_{L^{\infty}}$ and 1 are the Lipschitz constants of $\gamma '$ and γ, respectively. The first summand on the right-hand side of the above equals $|\int _{s_{n,i}}^{s}\int _{t}^{u} \gamma ''(v)\,\mathrm {d}v \,\mathrm {d}u |$, whereas the second is bounded by $\int _{s_{n,j}}^{t}\int _{s_{n,j}}^{u} |\gamma ''(v)| \,\mathrm{d} v \,\mathrm {d}u$, so that we can summarize the estimate

$$\begin{aligned} & \bigl\vert d(t,s)-d(s_{n,j},s_{n,i}) \bigr\vert \\ &\quad \stackrel {}{\leq }K \vert s_{n,i}-s \vert _{ \mathbb{R}/ L\mathbb{Z}} \max \bigl\{ \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} \\ &\qquad{}+ K \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2} +2K \vert t- s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\quad \stackrel {}{\leq }2K \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &\qquad{}\times \bigl[\max \bigl\{ \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t- s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} + \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} + \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr] \\ &\quad \stackrel {\text{(32)}}{\leq } 6 K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$

(37)

Inserting (31) and (37) into (29) yields

$$\begin{aligned} & \bigl\vert d^{q}(t,s)-d^{q}(s_{n,j}, s_{n,i}) \bigr\vert \\ &\quad\leq { 6qK^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$

(38)

In order to obtain an estimate for the denominator of $A_{i,j}$ we consider

$$\begin{aligned} \bigl\vert \gamma (s)-\gamma (t) \bigr\vert ^{2q} \stackrel {\text{(20)}}{\geq } \frac{1}{c_{\gamma}^{2q}} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q} \stackrel {\text{(27)}}{\geq } \biggl(\frac{c_{1}}{2c_{2} c_{\gamma}} \biggr)^{2q} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q}. \end{aligned}$$

(39)

Setting $C_{A}:= \frac{c_{2}^{3}}{c_{1}} 6 q K^{q} (1+2\frac{c_{2}}{c_{1}})^{2q-1} (\frac{2c_{2} c_{\gamma}}{c_{1}})^{2q}$ we obtain from (38) and (39)

$$\begin{aligned} \vert A_{i,j} \vert & \stackrel {\text{ (38)},\text{(39)}}{\leq } \frac{c_{1}}{ (c_{2} )^{3}} C_{A} \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}}\frac{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-1}\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert }{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q}}\,\mathrm {d}s \,\mathrm {d}t \\ &\stackrel {}{\leq }\frac{c_{1}}{ (c_{2} )^{3}} C_{A} \Bigl(\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ &\stackrel {\text{(17)}}{\leq } c_{1} C_{A} \frac{1}{n^{3}}\frac{1}{ \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert } \stackrel {\text{(17)}}{\leq } C_{A} \frac{1}{n^{2}} \frac{1}{ \vert i-j \vert _{n}}. \end{aligned}$$

(40)

Step 6: To estimate $B_{i,j}$, we use (28) and twice (27) leading to

$$\begin{aligned} & \bigl\vert \bigl|\gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{2q}- \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert ^{2q}\bigr| \\ &\quad \stackrel {}{=}\bigl\vert \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{q}+ \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert ^{q} \bigr\vert \\ &\qquad{}\times\bigl| \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{q}- \vert \gamma (s )-\gamma (t ) \vert ^{q}\bigr| \\ &\quad \stackrel {\text{(28)}}{\leq } q \bigl( \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{q}+ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{q} \bigr)\bigl|\bigl|\gamma (s_{n,i} )-\gamma (s_{n,j} )\bigr|-\bigl|\gamma (s )-\gamma (t ) \bigr\vert \bigr\vert \\ &\qquad{}\times \max \bigl\{ \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert , \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert \bigr\} ^{q-1} \\ &\quad \stackrel {\text{(27)}}{\leq } 2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{q}|s_{n,i}-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{q} \bigl\vert \gamma (s_{n,i} )-\gamma (s)+\gamma (t)-\gamma (s_{n,j} ) \bigr\vert \\ &\qquad{}\times \max \bigl\{ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} ^{q-1} \\ &\quad \stackrel {\text{(27)}}{\leq } 2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}|s_{n,i}-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \bigl( \bigl\vert \gamma (s )-\gamma (s_{n,i} ) \bigr\vert + \bigl\vert \gamma (t )-\gamma (s_{n,j} ) \bigr\vert \bigr) \\ &\quad \stackrel {}{\leq }2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1} |s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \bigl( \vert s-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}}+ \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr) \\ &\quad \stackrel {}{\leq }4q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}|s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$

(41)

Thus, by (30), (41), and (39),

$$\begin{aligned} \vert B_{i,j} \vert \stackrel {\text{(30)}}{\leq }{}& K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q} \\ &{}\times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \biggl\vert \frac{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}- \vert \gamma (s )-\gamma (t ) \vert ^{2q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q} \vert \gamma (s )- \gamma (t ) \vert ^{2q}} \biggr\vert \,\mathrm {d}s \,\mathrm {d}t \\ \stackrel {\text{(41)}}{\leq }{}& 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q-1}\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &{} \times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \frac{1}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q} \vert \gamma (s )-\gamma (t ) \vert ^{2q}} \,\mathrm {d}s \, \mathrm {d}t \\ \stackrel {\text{(39)}}{\leq }{}& \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &{} \times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q}} \,\mathrm {d}s \, \mathrm {d}t \\ \stackrel {}{\leq }{}& \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \Bigl( \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ \stackrel {\text{(17)}}{\leq }{}& (c_{2} )^{3} \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \frac{1}{n^{3}} \frac{1}{ \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert } \\ \stackrel {\text{(17)}}{\leq }{}&\frac{C_{B}}{n^{2}} \frac{1}{ \vert i-j \vert _{n}} \end{aligned}$$

(42)

with $C_{B}:=\frac{(c_{2})^{3}}{c_{1}} ( \frac{2c_{2} c_{\gamma}}{c_{1}} )^{2q}c_{\gamma}^{2q} 4q K^{q} (1+2\frac{c_{2}}{c_{1}} )^{4q-1}$.

Step 7: The expression $\sum_{k=1}^{n} \frac{1}{k}-\ln (n)$ converges for $n \to \infty$ to the Euler–Mascheroni constant; see [18, p. xix]. Thus, there exists a constant $c_{l}\in (0,\infty )$ such that $\vert \sum_{k=1}^{n} \frac{1}{k}-\ln (n) \vert \leq c_{l}$ for all $n \in \mathbb{N}$. This leads for $n \geq 1$ to

$$\begin{aligned} & \sum_{i=0}^{n-1} \sum _{j, \vert i-j \vert _{n}>\Upsilon} \bigl( \vert A_{i,j} \vert + \vert B_{i,j} \vert \bigr) \\ &\quad \stackrel {\text{(40)},\text{ (42)}}{\leq } \frac{2\max \{C_{A},C_{B}\}}{n^{2}}\sum_{i=0}^{n-1} \sum_{j, \vert i-j \vert _{n}>\Upsilon} \frac{1}{ \vert i-j \vert _{n}} \\ &\quad \stackrel {}{\leq }\frac{4\max \{C_{A},C_{B}\}}{n^{2}}\sum_{i=0}^{n-1} \sum_{k=1}^{n} \frac{1}{k} \\ &\quad \stackrel {}{=}\frac{4\max \{C_{A},C_{B}\}}{n} \Biggl(\sum_{k=1}^{n} \frac{1}{k}-\ln (n) \Biggr)+4\max \{C_{A},C_{B}\} \frac{\ln (n)}{n} \stackrel {}{\leq }\frac{C_{AB}\ln (n)}{n} \end{aligned}$$

(43)

with $C_{AB}:=8 \max \{c_{l},1\}\max \{C_{A},C_{B}\}$.

Step 8: Recall from Step 3 that

$$\begin{aligned} { \bigl\vert \lambda _{n,j}- \vert s_{n,j+1}- s_{n,j} \vert \bigr\vert \leq c_{K} \vert s_{n,j+1}- s_{n,j} \vert ^{3}} \leq c_{K} \Bigl(\max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \end{aligned}$$

(44)

holds for all $n \geq N$ and $j=0,\dots, n-1$, where N depends on the sequence $(\beta _{n})_{n \in \mathbb{N}}$. From (22) and (24) we obtain from (44)

$$\begin{aligned} \vert C_{i,j} \vert \stackrel {}{\leq }{} & \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,i}\lambda _{n,j}\bigr\vert \\ \stackrel {\text{(22)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \bigl\vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,j} \bigr\vert \\ &{}+\lambda _{n,j} \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert -\lambda _{n,i} \bigr\vert \bigr] \\ \stackrel {\text{(24)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \bigl\vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,j} \bigr\vert \\ &{}+ \vert s_{n,j+1}-s_{n,j} \vert \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert -\lambda _{n,i} \bigr\vert \bigr] \\ \stackrel {\text{(44)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} c_{K} \Bigl(\max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \\ &{}\times \bigl[ \vert s_{n,i+1}-s_{n,i} \vert + \vert s_{n,j+1}- s_{n,j} \vert \bigr] \\ \stackrel {}{\leq }{}& 2 \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} c_{K} \Bigl(\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{4} \stackrel {\text{ (17)}}{\leq }\frac{C_{C}}{n^{4}}, \end{aligned}$$

(45)

with $C_{C}:=2 (c_{2} )^{4} (c_{\gamma}^{2} K )^{q} d_{K} c_{K}$ for all $n \geq N$. We then conclude that

$$\begin{aligned} \sum_{i=0}^{n-1} \sum _{j, \vert i-j \vert _{n}>\Upsilon} \vert C_{i,j} \vert \stackrel {\text{ (45)}}{\leq } \frac{C_{C}}{n^{4}}\sum_{i=0}^{n-1} \sum_{j=0}^{n-1} 1 = \frac{C_{C}}{n^{2}}. \end{aligned}$$

(46)

Step 9: Inserting (23), (25), (43), and (46) into (19) yields

$$\begin{aligned} \bigl\vert \mathrm {TP}_{q} (\gamma )-\tilde{\mathcal {E}}_{q}^{n} (\beta _{n} ) \bigr\vert \leq & \frac{C_{1}}{n}+ \frac{C_{2}}{n}+2^{q} \biggl( \frac{C_{AB} \ln (n)}{n} + \frac{C_{C}}{n^{2}} \biggr) \leq \frac{C \ln (n)}{n} \quad\text{for } n \geq N, \end{aligned}$$

with $C:=4\max \{C_{1},C_{2},2^{q} C_{AB}, 2^{q} C_{C}\}$, which gives the desired result. □

Γ-convergence to the continuous tangent-point energy

In the present section, we show that the continuous tangent-point energy $\mathrm {TP}_{q}$ is the Γ-limit of the discrete tangent-point energies $\mathcal {E}_{q}^{n}$ as $n\to \infty$ (see Theorem 1.1). As a consequence, we deduce that the limits of discrete almost minimizers are minimizers of the continuous tangent-point energy; see Corollary 1.2.

Γ-convergence

In order to prove Theorem 1.1 we need to verify the liminf and limsup inequalities, see [8, Definition 1.5]. Here, the liminf inequality is verified in a rather straightforward manner (Theorem 4.1), whereas the proof of the limsup inequality requires more work; see Theorem 4.4 below. Similarly as the notation $C_{ia}$ used before, we equip a function space $\mathcal{S}$ with the index a if we take arclength-parametrized curves in that space.

Theorem 4.1

(Liminf inequality)

Let $\gamma, \gamma _{n} \in C^{1}_{\mathrm{ {a}}} ( \mathbb{R}/ L\mathbb{Z}, \mathbb{R}^{3} )$ with $\gamma _{n} \stackrel{C^{1}}{\longrightarrow} \gamma$ as $n \to \infty$. Then, $\mathrm {TP}_{q}(\gamma )\leq \liminf_{n \to \infty} \mathcal {E}_{q}^{n}( \gamma _{n})$.

Proof

We may assume that $\liminf_{n \to \infty} \mathcal {E}_{q}^{n}(\gamma _{n}) < \infty$. Then, there exists a subsequence $(\gamma _{n_{k}} )_{k \in \mathbb{N}}$ satisfying $\liminf_{n \to \infty} \mathcal {E}_{q}^{n}(\gamma _{n})=\lim_{k \to \infty} \mathcal {E}_{q}^{n_{k}} (\gamma _{n_{k}} )<\infty$. By definition of $\mathcal {E}_{q}^{n_{k}}$ we deduce $\gamma _{n_{k}} \in \mathcal {B}_{n_{k}}$ for all $k \in \mathbb{N}$; see (8) in the introduction. Denote the point-tangent pairs that are interpolated by $\gamma _{n_{k}}$ as $( ( [q_{{n_{k}},i},t_{{n_{k}},i} ], [q_{{n_{k}},i+1},t_{{n_{k}},i+1} ] ) )_{i=0, \dots, {n_{k}}-1}$, with $q_{{n_{k}},0}=q_{{n_{k}},{n_{k}}}$ and $t_{{n_{k}},0}=t_{{n_{k}},{n_{k}}}$ for each $k\in \mathbb{N}$. Furthermore, we denote by $a_{{n_{k}},0}, \dots, a_{{n_{k}},{n_{k}}}$ the arclength parameters satisfying $\gamma _{n_{k}}(a_{{n_{k}},i})=q_{{n_{k}},i}$ and $|a_{{n_{k}},i+1}-a_{{n_{k}},i}|=\lambda _{{n_{k}},i}$ for all $i=0, \dots, {n_{k}}-1$. Define for all $s,t \in \mathbb{R}/ L\mathbb{Z}$ with $s\neq t$ the function

$$\begin{aligned} f_{n_{k}}(s,t):= \sum_{i=0}^{{n_{k}}-1} \sum_{j=0, j\neq i}^{{n_{k}}-1} { \biggl(2 \frac{\operatorname{dist}(l(\gamma _{n_{k}}(a_{{n_{k}},j})),\gamma _{n_{k}}(a_{{n_{k}},i}) )}{ \vert \gamma _{n_{k}}(a_{{n_{k}},i})-\gamma _{n_{k}}(a_{{n_{k}},j}) \vert ^{2}} \biggr)^{q}} \chi _{[a_{{n_{k}},i},a_{{n_{k}},i+1})\times [a_{{n_{k}},j},a_{{n_{k}},j+1})}(s,t), \end{aligned}$$

where $\chi _{A}$ denotes the characteristic function of a set $A\subset \mathbb{R}/L\mathbb{Z}\times \mathbb{R}/L\mathbb{Z}$. Easy calculations show that

$$\begin{aligned} \lim_{k \to \infty} f_{n_{k}}(s,t)= \biggl( \frac{2\operatorname{dist}(l(\gamma (t)),\gamma (s))}{ \vert \gamma (s)-\gamma (t) \vert ^{2}} \biggr)^{q} \quad \text{for any }s\neq t. \end{aligned}$$

The functions $f_{n_{k}}$ are nonnegative and measurable since they are piecewise constant. We can rewrite the discrete tangent-point energies as $\mathcal {E}^{n}_{q}(\gamma _{n_{k}})= \int _{\mathbb{R}/L\mathbb{Z}} \int _{\mathbb{R}/L\mathbb{Z}}f_{n_{k}}(s,t)\,\mathrm {d}s \,\mathrm {d}t$, which allows us to apply Fatou’s lemma to obtain the desired liminf inequality. □

An important first ingredient in the proof of the limsup inequality is the use of convolutions

$$\begin{aligned} \gamma _{\varepsilon}(x):= (\gamma \ast \eta _{\varepsilon} ) (x)= \int _{\mathbb{R}}\gamma (x-y)\eta _{\varepsilon}(y)\,\mathrm {d}y\quad \text{for }x\in \mathbb{R}/ L\mathbb{Z} \end{aligned}$$

(47)

that approximate γ in the $C^{1}$-norm. Here, $\eta \in C^{\infty} (\mathbb{R})$ is a nonnegative mollifier with $\operatorname{supp}\eta \subset [-1,1]$ and $\int _{\mathbb{R}} \eta (x)\,\mathrm {d}x =1$, and for any $\varepsilon >0$ we set $\eta _{\varepsilon}(x):=\frac{1}{\varepsilon}\eta ( \frac{x}{\varepsilon} )$.

In general, the convolutions are not parametrized by arclength even if γ is, and they do not need to have the same length as γ. Thus, we rescale the convolutions to have the same length as γ and reparametrize then according to arclength. The following theorem extends [6, Theorem 1.3] to the case $\rho \ge \frac{1}{s}$. A proof can be found in Appendix A.

Theorem 4.2

Let $s\in (0,1)$, $\rho \in [\frac{1}{s},\infty )$, and $\gamma \in W_{\mathrm{ {a}}}^{1+s,\rho} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$. For $\varepsilon >0$ denote by $\tilde{\gamma}_{\varepsilon}$ be the arclength parametrization of the rescaled convolutions $\frac{\mathscr{L} (\gamma )}{\mathscr{L} (\gamma _{\varepsilon} )} \gamma _{\varepsilon}$ with $\tilde{\gamma}_{\varepsilon}(0)= \frac{\mathscr{L} (\gamma )}{\mathscr{L} (\gamma _{\varepsilon} )} \gamma _{\varepsilon}(0)$, where $\mathscr{L} (\gamma )$ is the length of γ and $\mathscr{L} (\gamma _{\varepsilon} )$ is the length of $\gamma _{\varepsilon}$. Then, $\tilde{\gamma}_{\varepsilon }\to \gamma$ in $W^{1+s,\rho}$ as $\varepsilon \to 0$.

The following abstract lemma is a specialization of [19, Lemma 6.1.1] and provides sufficient conditions to transfer the limsup inequality from approximating elements to the limit element. This result applied to smooth convolutions approximating a given $C^{1}$-curve γ will be the second ingredient in the proof of the limsup inequality, Theorem 4.4 below.

Lemma 4.3

(Limsup inequality by approximation)

Let $(X,d)$ be a metric space and $\mathcal{F}_{n}, \mathcal{F}: X \to [-\infty,\infty ]$. If a sequence $(x^{m} )_{m \in \mathbb{N}} \subset X$ satisfies

• 1. $d (x,x^{m} ) \to 0$ as $m \to \infty$ for an element $x \in X$;

• 2. $\limsup_{m \to \infty} \mathcal{F} (x^{m} )\leq \mathcal{F}(x)$;

• 3. for every $m \in \mathbb{N}$ there exists a sequence $(x^{m}_{n} )_{n \in \mathbb{N}}$ with $d (x^{m},x^{m}_{n} ) \to 0$ as $n \to \infty$ and $\limsup_{n \to \infty} \mathcal{F}_{n} (x^{m}_{n} )\leq \mathcal{F} (x^{m} )$,

then there exists a sequence $(y_{n} )_{n \in \mathbb{N}}\subset X$ with

$$\begin{aligned} d (x,y_{n} ) \to 0\quad \textit{as } n \to \infty\quad \textit{and}\quad \limsup _{n \to \infty} \mathcal{F}_{n} (y_{n} ) \leq \mathcal{F}(x). \end{aligned}$$

The proof of the limsup inequality is inspired by Blatt’s improvement of Scholtes’ Γ-convergence result for the Möbius energy [6, Theorem 4.8].

Theorem 4.4

(Limsup inequality)

For every $\gamma \in C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ there exists a sequence $(b_{n} )_{n \in \mathbb{N}} \subset C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ such that

$$\begin{aligned} b_{n} \stackrel{C^{1}}{\underset{n \to \infty}{ \longrightarrow}} \gamma \quad\textit{and}\quad \limsup_{n \to \infty} \mathcal {E}_{q}^{n} (b_{n} ) \leq \mathrm {TP}_{q}( \gamma ). \end{aligned}$$

Proof

If $\mathrm {TP}_{q}(\gamma )=\infty$, choose $b_{n}=\gamma$ for all $n \in \mathbb{N}$. Then, $b_{n}\to \gamma$ in $C^{1}$ and the limsup inequality follows trivially. From now on let $\mathrm {TP}_{q}(\gamma )<\infty$. Thus, we have $\gamma \in W^{2-\frac{1}{q},q} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ by [5, Theorem 1.1]. Moreover, Lemma B.1 yields a $c_{\gamma }>0$ such that

$$\begin{aligned} c_{\gamma } \bigl\vert \gamma (s)-\gamma (t) \bigr\vert \geq \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}\quad \text{for any }t,s\in \mathbb{R}. \end{aligned}$$

(48)

We now consider a sequence of suitably rescaled and reparametrized convolutions of γ and prove the limsup inequality for these convolutions. Applying Lemma 4.3 then yields the limsup inequality for γ.

Step 1: For $n \in \mathbb{N}$ define $s_{n,i}:=\frac{iL}{n}$ for $i=0, \dots, n$. Then, for all $i=0, \dots,n-1$ we have $\vert s_{n,i+1}-s_{n,i} \vert =h_{n}=\tilde{h}_{n}=\frac{L}{n}$, so that the $\mathcal{M}_{n}:=\{s_{n,0},\ldots, s_{n,n}\}$ form a $(c_{1}-c_{2})$-distributed sequence of partitions with $c_{1}=c_{2}=L$ for $n\ge 2$; see Definition 2.8.

Step 2: For $k \in \mathbb{N}$ we set $\varepsilon _{k}:=\frac{1}{k}$. Let $\gamma _{\varepsilon _{k}}$ be the convolution as in (47) and $\mathscr{L} (\gamma _{\varepsilon _{k}} )$ the length of $\gamma _{\varepsilon _{k}}$. We then define $\tilde{\gamma}_{k}$ as the arclength parametrization of the rescalings $L\gamma _{\varepsilon _{k}}/\mathscr{L} (\gamma _{\varepsilon _{k}} )$. Thus, $\tilde{\gamma}_{k}$ has the same length as γ for every $k \in \mathbb{N}$. Furthermore, $\tilde{\gamma}_{k}$ is on $[0,L)$ injective for k sufficiently large, which follows from the bi-Lipschitz property (48) of γ together with the $C^{1}$-convergence of the convolutions $\gamma _{\varepsilon _{k}}\to \gamma$ as $k\to \infty$. By omitting finitely many indices we may assume that $\tilde{\gamma }_{k} \in C^{\infty}_{\mathrm{ {ia}}}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ for all $k\in \mathbb{N}$. For every $k\in \mathbb{N}$ there is by Lemma 2.9 some index $N_{0}(k)\in \mathbb{N}$ such that there exist proper $\tilde{\gamma}_{k}$-interpolating balanced biarc curves $\tilde{\beta}_{n}^{k}$ parametrized by arclength that interpolate the point-tangent pairs $( ( [\tilde{\gamma}_{k} (s_{n,i} ), \tilde{\gamma}_{k}' (s_{n,i} ) ], [ \tilde{\gamma}_{k} (s_{n,i+1} ),\tilde{\gamma}_{k}' (s_{n,i+1} ) ] ) )_{i=0, \dots, n-1}$ for all $n\ge N_{0}(k)$, such that the matching points $m_{n,i}^{k} \in \Sigma _{++}^{n,i,k}$ satisfy (see Definitions 2.7 and 2.5(iv)) $\vert \tilde{\gamma}_{k} (s_{n,i} )-m_{n,i}^{k} \vert = \vert \tilde{\gamma}_{k} (s_{n,i+1} )-m_{n,i}^{k} \vert$ for all $n\ge N_{0}(k)$, $i=0,\ldots,n-1$. Let $L_{n}^{k}:=\mathscr{L}(\tilde{\beta}_{n}^{k})$ be the length of $\tilde{\beta}_{n}^{k}$, and note that Theorem 2.11 implies

$$\begin{aligned} L_{n}^{k}\to L=\mathscr{L}(\tilde{ \gamma }_{k}) \quad\text{{for each $k\in \mathbb{N}$ as $n\to \infty $.}} \end{aligned}$$

(49)

Step 3: For $k \in \mathbb{N}$, let $\varphi _{n}^{k}$ be Smutny’s reparametrization [31, Appendix A] and define $\tilde{B}_{n}^{k}:=\tilde{\beta}_{n}^{k}\circ \varphi _{n}^{k}$, so that Theorem 2.12 implies

$$\begin{aligned} \bigl\Vert \tilde{\gamma}_{k}-\tilde{B}_{n}^{k} \bigr\Vert _{C^{1}} \to 0 \quad\text{{ for each $k\in \mathbb{N}$ as $ n \to \infty $.}} \end{aligned}$$

(50)

Now, define $B_{n}^{k}(s):=L (L_{n}^{k} )^{-1} \tilde{B}_{n}^{k}(s)$ for all $s \in \mathbb{R}$. Then, $B_{n}^{k}$ obviously has length L. However, $B_{n}^{k}$ is not parametrized by arclength. Nevertheless, by means of (49) we find $\|B^{k}_{n}-\tilde{B}^{k}_{n}\|_{C^{1}}= | \frac{L}{L^{k}_{n}}-1 |\|\tilde{B}^{k}_{n}\|_{C^{1}}\to 0$ for each $k\in \mathbb{N}$ as $n\to \infty$. Consequently, by (50), one has $\|\tilde{\gamma }_{k}-B^{k}_{n}\|_{C^{1}}\to 0$ for each $k\in \mathbb{N}$ as $n\to \infty$, and therefore by means of Lemma B.3,

$$\begin{aligned} \bigl\Vert \tilde{\gamma}_{k} - \beta _{n}^{k} \bigr\Vert _{C^{1}} \longrightarrow 0\quad \text{{for each $k\in \mathbb{N}$ as $n\to \infty $,}} \end{aligned}$$

(51)

where $\beta _{n}^{k}$ is the reparametrization of $B_{n}^{k}$ by arclength. Again, since $\tilde{\gamma }_{k}$ is injective for k sufficiently large, this implies that $\beta _{n}^{k} \in C_{\mathrm{ {ia}}}^{1}( \mathbb{R}/ \mathbb{Z},\mathbb{R}^{3})$ for n and k sufficiently large.

Step 4: We now show that $\beta _{n}^{k} \in \mathcal{B}_{n}$ holds if n is sufficiently large, such that the values $\mathcal {E}_{q}^{n}(\beta _{n}^{k})$ are finite by definition (8) in the introduction. We need to show that the length $\lambda _{n,i}^{k}$ of the ith biarc of $\beta _{n}^{k}$ satisfies (7). For that we apply Lemma 2.10 to the length $\tilde{\lambda}_{n,i}^{k}$ of the ith biarc of $\tilde{\beta}_{n}^{k}$. More precisely, we take the limit $n\to \infty$ in the following inequality that holds for each $k\in \mathbb{N}$, $n\ge N_{0}(k)$, $i=0,\ldots,n-1$,

$$\begin{aligned} -\max_{j=0, \dots, n-1} \biggl\vert { \frac{\tilde{\lambda}_{n,j}^{k}}{({L}/{n})} -1} \biggr\vert + 1\le { \frac{\tilde{\lambda}_{n,i}^{k}}{({L}/{n})}} &\leq \max_{j=0, \dots, n-1} \biggl\vert { \frac{\tilde{\lambda}_{n,j}^{k}}{({L}/{n})}} -1 \biggr\vert + 1, \end{aligned}$$

to obtain

$$\begin{aligned} 1\longleftarrow \min_{i=0, \dots, n-1} { \frac{\tilde{\lambda}_{n,i}^{k}}{({L}/{n})}}\le \max _{i=0, \dots, n-1} \frac{\tilde{\lambda}_{n,i}^{k}}{({L}/{n})} \longrightarrow 1 \quad\text{{as $n\to \infty $.}} \end{aligned}$$

(52)

Since the image of $\beta _{n}^{k}$ is just the image of $\tilde{\beta}_{n}^{k}$ scaled by the factor $L (L_{n}^{k} )^{-1}$ we deduce $\lambda _{n,j}^{k}=L (L_{n}^{k} )^{-1} \tilde{\lambda}_{n,j}^{k}$. Combining this with (49) we find for each $k\in \mathbb{N}$ and index $N_{1}(k)\ge N_{0}(k)$ such that

$$\begin{aligned} { \frac{L}{2n}} \leq \min_{j=0, \dots,n-1}\lambda _{n,j}^{k} \leq \max_{j=0, \dots,n-1}\lambda _{n,j}^{k} \leq { \frac{2L}{n}} \quad \text{for any }n\ge N_{1}(k), \end{aligned}$$

(53)

which is (7) for $\lambda _{i}:=\lambda _{i}^{k}$. Thus, $\beta _{n}^{k} \in \mathcal{B}_{n}$ for all $n\ge N_{1}(k)$.

Step 5: The scaling property (9) and the parameter invariance of the discrete tangent-point energies yields

$$\begin{aligned} \mathcal {E}_{q}^{n} \bigl(\beta _{n}^{k} \bigr) = \bigl(L \bigl(L_{n}^{k} \bigr)^{-1} \bigr)^{2-q} \tilde{\mathcal {E}}_{q}^{n} \bigl(\tilde{ \beta}_{n}^{k} \bigr) \quad\text{{for all $k\in \mathbb{N}$ and $n\ge N_{1}(k)$,}} \end{aligned}$$

so that we obtain by (49) and Theorem 3.1 applied to $\gamma :=\tilde{\gamma }_{k}$ and $\beta _{n}:=\tilde{\beta}^{k}_{n}$

$$\begin{aligned} \bigl\vert \mathrm {TP}_{q}(\tilde{\gamma}_{k})-\mathcal {E}_{q}^{n} \bigl(\beta _{n}^{k} \bigr) \bigr\vert \leq \bigl\vert \mathrm {TP}_{q}(\tilde{ \gamma}_{k}) - \tilde{\mathcal {E}}_{q}^{n} \bigl( \tilde{\beta}_{n}^{k} \bigr) \bigr\vert + \bigl\vert \tilde{\mathcal {E}}_{q}^{n} \bigl(\tilde{\beta}_{n}^{k} \bigr) \bigr\vert \bigl\vert 1- \bigl(L \bigl(L_{n}^{k} \bigr)^{-1} \bigr)^{2-q} \bigr\vert \longrightarrow 0 \end{aligned}$$

(54)

for each $k\in \mathbb{N}$ as $n\to \infty$.

Step 6: In this final step we check the assumptions of Lemma 4.3. The space $C_{\mathrm{ {ia}}}^{1} ( \mathbb{R}/ L\mathbb{Z}, \mathbb{R}^{3} )$ is a metric space with the metric induced by the $C^{1}$-norm. By Morrey–Sobolev embedding (see [21, Theorem A.2] in the setting of periodic functions) there exists a constant $c_{E} >0$, such that

$$\begin{aligned} \Vert \tilde{\gamma}_{k} - \gamma \Vert _{C^{1}}\leq c_{E} \Vert \tilde{\gamma}_{k} - \gamma \Vert _{W^{2-\frac{1}{q},q}}. \end{aligned}$$

According to Theorem 4.2 applied to $\rho =q$ and $s=1-\frac {1}{q}$ for $q>2$ the right-hand side converges to 0 as $k \to \infty$. Thus, $\tilde{\gamma}_{k}$ converges in the $C^{1}$-norm to γ, which verifies condition (i) in Lemma 4.3. Furthermore, [34, (4.2) Satz] implies that $\mathrm {TP}_{q}$ is continuous on $W_{\mathrm{ {ia}}}^{2-\frac{1}{q},q}$ since $q > 2$. Thus, we obtain $\lim_{k \to \infty} \mathrm {TP}_{q} (\tilde{\gamma}_{k} ) = \mathrm {TP}_{q}(\gamma )$, which gives us condition (ii) of Lemma 4.3. Combining (51) with (54) verifies condition (iii) of Lemma 4.3. Hence, Lemma 4.3 yields the limsup inequality for γ. □

Remark 4.5

For the proof of Theorem 1.3 in Sect. 5 (see in particular Lemma 5.2) it is important to note that the actual recovery sequence for the limsup inequality in the previous proof is a subsequence of the (doubly subscripted) arclength parametrized biarc curves $\beta ^{k}_{n}\in C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ for $k\in \mathbb{N}$ and $n\ge N_{1}(k)$; see the choice of the abstract recovery sequence towards the end of the proof of [19, Lemma 6.1.1].

Proof of Theorem 1.1

According to [8, Definition 1.5] it suffices to verify two fundamental inequalities. Indeed, the liminf inequality is the content of Theorem 4.1, whereas the limsup inequality is established in Theorem 4.4. □

Convergence of discrete almost minimizers

In this subsection, we prove the convergence of discrete almost minimizers of the discrete tangent-point energies in the metric space defined before. The following lemma can be found in [12, Corollary 7.20].

Lemma 4.6

(Convergence of minimizers)

Let $(X,d)$ be a metric space and $\mathcal{F}_{n}, \mathcal{F}: X \to [-\infty,\infty ]$. Assume that $\mathcal{F}_{n} \stackrel{\Gamma}{\underset{n \to \infty}{\longrightarrow}} \mathcal{F}$. Let $\vert \mathcal{F}_{n}(z_{n}) - \inf_{X} \mathcal{F}_{n} \vert \to 0$ and $z_{n} \to z \in X$ as n tends to infinity. Then, $\mathcal{F}(z) = \min_{X} \mathcal{F}$ and $\lim_{n \to \infty} \mathcal{F}_{n}(z_{n})=\mathcal{F}(z)$.

Proof of Corollary 1.2.

The proof follows immediately from Lemma 4.6 with the metric space $X=C_{\mathrm{ {ia}}}^{1} ( \mathbb{R}/ L\mathbb{Z}, \mathbb{R}^{3} )\cap \mathcal{K}$, with the metric induced by the $C^{1}$-norm. Note that the knot class $\mathcal{K}$ is stable under $C^{1}$-convergence; see, e.g., [25]. Since $\mathrm {TP}_{q}(\gamma )<\infty$ holds, we obtain $\gamma \in W^{2-\frac{1}{q},q} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ by [5, Theorem 1.1]. □

Γ convergence to the Ropelength functional

As a first step towards the proof of Theorem 1.3 we show that the continuous tangent-point energies $(\mathrm {TP}_{k})^{\frac{1}{k}}$ Γ-converge to the ropelength $\mathcal{R}$ on $C_{\mathrm{ {ia}}}^{1,1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3})$ equipped with the $C^{1}$-norm as $k\to \infty$. We follow the proof of [14, Theorem 6.11], where Gilsbach showed Γ-convergence of integral Menger curvatures towards ropelength.

Lemma 5.1

For any $\gamma \in C^{1,1}_{\mathrm{ {ia}}}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3})$ one has $(\mathrm {TP}_{k})^{\frac{1}{k}}(\gamma )\to \mathcal {R}(\gamma )$ as $k\to \infty$. Moreover, $(\mathrm {TP}_{k})^{\frac{1}{k}} \stackrel{\Gamma}{\underset{k \to \infty}{\longrightarrow}} \mathcal {R}$ on $(C_{\mathrm{ {ia}}}^{1,1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3}),\|\cdot \|_{C^{1}} )$.

Proof

According to [30, Theorem 1(iii)] one has³$\mathcal {R}(\gamma )<\infty$ for $\gamma \in C_{\mathrm{ {ia}}}^{1,1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3})$. In addition, by [30, Lemma 2]

$$\begin{aligned} \bigl\Vert r^{-1}_{\mathrm{ {tp}}} \bigl(\gamma (\cdot ),\gamma ( \cdot ) \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z})}= \sup_{s,t \in \mathbb{R}/\mathbb{Z}, s\neq t} r^{-1}_{\mathrm{ {tp}}} \bigl(\gamma (s),\gamma (t) \bigr) = \frac{1}{\triangle [\gamma ]}=\mathcal {R}(\gamma ). \end{aligned}$$

(55)

It is well known (see, e.g., [1, E3.4]) that the mapping

$$\begin{aligned} k\mapsto \bigl\Vert r^{-1}_{\mathrm{ {tp}}} \bigl(\gamma (\cdot ), \gamma (\cdot ) \bigr) \bigr\Vert _{L^{k}(\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z})}=(\mathrm {TP}_{k})^{ \frac{1}{k}} \end{aligned}$$

is nondecreasing and satisfies by means of (55)

$$\begin{aligned} \lim_{k\to \infty}(\mathrm {TP}_{k})^{\frac{1}{k}}(\gamma ) = \bigl\Vert r^{-1}_{ \mathrm{{tp}}} \bigl(\gamma (\cdot ),\gamma (\cdot ) \bigr) \bigr\Vert _{L^{\infty}( \mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z})}\overset{\text{(55)}}{=} \mathcal {R}(\gamma ). \end{aligned}$$

Furthermore, the continuous tangent-point energy is lower semicontinuous with respect to the $C^{1}$-norm, see [32, Proof of Corollary 2.3] or [14, Lemma 1.41]. Then, by [8, Remark 1.40(ii)] the pointwise limit of $(\mathrm {TP}_{k})^{\frac{1}{k}}$ is also the Γ-limit and we obtain $(\mathrm {TP}_{k})^{\frac{1}{k}} \stackrel{\Gamma}{\underset{}{\longrightarrow}} \mathcal {R}$ as $k\to \infty$. □

Lemma 5.2

$(\mathcal {E}_{q}^{n})^{\frac{1}{q}} \stackrel{\Gamma}{\underset{n \to \infty}{\longrightarrow}} (\mathrm {TP}_{q})^{ \frac{1}{q}}$ on $(C_{\mathrm{ {ia}}}^{1,1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3}), \|\cdot \|_{C^{1}} )$ for all $q > 2$.

Proof

By Theorem 1.1 we have $\mathcal {E}_{q}^{n} \stackrel{\Gamma}{\underset{}{\longrightarrow}} \mathrm {TP}_{q}$ on $(C_{\mathrm{ {ia}}}^{1}( \mathbb{R}/ \mathbb{Z}, \mathbb{R}^{3}), \|\cdot \|_{C^{1}})$ for any $q>2$ as $n\to \infty$. However, in the proof of the limsup inequality in Theorem 4.4 the recovery sequence is a sequence consisting only of biarc curves that are in $C_{\mathrm{ {ia}}}^{1,1} ( \mathbb{R}/ \mathbb{Z}, \mathbb{R}^{3} )$; see Remark 4.5. Therefore, we also have $\mathcal {E}^{n}_{q} \stackrel{\Gamma}{\underset{}{\longrightarrow}} \mathrm {TP}_{q}$ on the space $(C_{\mathrm{ {ia}}}^{1,1}( \mathbb{R}/ \mathbb{Z}, \mathbb{R}^{3} ), \|\cdot \|_{C^{1}})$ as $n\to \infty$. Now, apply [12, Proposition 6.16] to $\mathcal{F}_{n}:=\mathcal {E}_{q}^{n}$, $\mathcal{F}:=\mathrm {TP}_{q}$ and the continuous and nondecreasing function $g: (0,\infty ) \to \mathbb{R}, x \mapsto x^{\frac{1}{q}}$ to infer $(\mathcal {E}_{q}^{n})^{\frac{1}{q}} = g \circ \mathcal{F}_{n} \stackrel{\Gamma}{\underset{n \to \infty}{\longrightarrow}} g\circ \mathcal{F} = (\mathrm {TP}_{q})^{\frac{1}{q}}$. □

Next, we compare two different discrete tangent-point energies.

Lemma 5.3

Let $n,m,k \in \mathbb{N}$, $2\le k \leq m$ and $\gamma \in C^{1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{3})$ with length $\mathscr{L}(\gamma )$. Then, $(\mathcal {E}_{k}^{n} )^{\frac{1}{k}}(\gamma ) \leq (\frac{4\mathscr{L} (\gamma )^{2}n(n-1)}{n^{2}} )^{\frac{1}{k}-\frac{1}{m}} ( \mathcal {E}_{m}^{n} )^{\frac{1}{m}}(\gamma )$.

Proof

We only have to consider the case that $\gamma \in \mathcal{B}_{n}$ since otherwise both sides of the inequality are infinite by definition of the discrete energy $\mathcal {E}_{k}^{n}$; see (8) in the introduction. Denote by $( ( [q_{i},t_{i} ], [q_{i+1},t_{i+1} ] ) )_{i=0,.., n-1}$ the point-tangent pairs that γ interpolates. For $i\neq j$ define $x_{i,j}:= \frac{2 \operatorname{dist}(l(q_{i}),q_{j})}{|q_{i}-q_{j}|^{2}}\geq 0$. Then, we estimate by means of the generalized mean inequality for finite sums, $(\frac{1}{\ell}\sum_{i=1}^{\ell}|a_{i}|^{p})^{\frac{1}{p}}\le ( \frac{1}{\ell}\sum_{i=1}^{\ell}|a_{i}|^{q})^{\frac{1}{q}}$ for $p\le q$ (here, for $\ell:=n(n-1)$, $p:=k$, $q:=m$), and by (7)

$$\begin{aligned} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\gamma )& \stackrel {}{=}\Biggl(\sum_{i=0}^{n-1} \sum _{j=0,j\neq i}^{n-1} x_{i,j}^{k} \lambda _{i} \lambda _{j} \Biggr)^{\frac{1}{k}}\\ &\stackrel {}{=}\Biggl(\sum _{i=0}^{n-1} \sum _{j=0,j\neq i}^{n-1} \bigl[x_{i,j} (\lambda _{i} \lambda _{j})^{ \frac{1}{k}} \bigr]^{k} \Biggr)^{\frac{1}{k}} \\ & \stackrel {}{\leq }\bigl(n(n-1) \bigr)^{\frac{1}{k}-\frac{1}{m}} \Biggl(\sum _{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} x_{i,j}^{m} \underbrace{ (\lambda _{i} \lambda _{j} )^{\frac{m}{k}}}_{= (\lambda _{i} \lambda _{j} ) (\lambda _{i} \lambda _{j} )^{\frac{m}{k}-1}} \Biggr)^{\frac{1}{m}} \\ &\stackrel {}{\leq }\bigl(n(n -1) \bigr)^{\frac{1}{k}-\frac{1}{m}} \max_{ \substack{i,j=0,\dots,n-1 \\ i \neq j}} \underbrace{(\lambda _{i} \lambda _{j})^{\frac{1}{k}-\frac{1}{m}}}_{ \overset{\text{(7)}}{\leq} \left( \frac{4\mathscr{L}(\gamma )^{2}}{n^{2}} \right)^{\frac{1}{k}- \frac{1}{m}}} \underbrace{ \Biggl(\sum_{i=0}^{n-1} \sum _{j=0,j\neq i}^{n-1} x_{i,j}^{m} \lambda _{i} \lambda _{j} \Biggr)^{\frac{1}{m}}}_{= (\mathcal {E}_{m}^{n} )^{\frac{1}{m}}(\gamma )}. \end{aligned}$$

 □

Proof of Theorem 1.3

It suffices to prove the Γ-convergence for $L=1$, since then the statement for general L follows from the scaling property and parametrization invariance of the energies involved. Indeed, assume the theorem was proven for $L=1$. Now, take $L\neq 1$ and let $(\gamma _{n})_{n\in \mathbb{N}}\subset C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ with $\gamma _{n}\to \gamma$ in $C^{1}$ as $n\to \infty$. Denote by $\tilde{\gamma }_{n}$ the arclength parametrization of $\frac{\gamma _{n}}{L}$. By Lemma B.3 this implies $\tilde{\gamma }_{n} \to \tilde{\gamma }$ in $C^{1}$ as $n \to \infty$, where γ̃ is the arclength parametrization of $\frac{\gamma }{L}$. Together with the fact that the ropelength functional is invariant under reparametrization and scaling, the liminf inequality for $L=1$ yields the liminf equality for general L:

$$\begin{aligned} \mathcal {R}(\gamma )=\mathcal {R}(\tilde{\gamma })\leq \liminf_{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\tilde{ \gamma }_{n}) =\liminf_{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}{ \biggl(\frac{\gamma _{n}}{L} \biggr)} \overset{\text{(9)}}{=} \liminf_{n \to \infty}L^{\frac{n-2}{n}} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}} (\gamma _{n} ). \end{aligned}$$

For the limsup inequality let $\gamma \in C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$. Then, $\tilde{\gamma }(x):=\frac{\gamma (Lx)}{L}$ is the arclength parametrization of γ scaled to unit length. Hence, there exists a recovery sequence $(\tilde{\gamma }_{n})_{n\in \mathbb{N}}\subset C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ \mathbb{Z},\mathbb{R}^{3})$ such that

$$\begin{aligned} \tilde{\gamma }_{n} \stackrel{C^{1}}{\to} \tilde{\gamma }\quad \text{as } n \to \infty\quad \text{and}\quad \limsup_{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\tilde{\gamma }_{n}) \leq \mathcal {R}(\tilde{\gamma }). \end{aligned}$$

(56)

Define the reparametrization $\varphi: [0,L] \to [0,1], x \mapsto \frac{x}{L}$ and set $\gamma _{n}(x):= L \tilde{\gamma _{n}}(\varphi (x))$ and $\hat{\gamma }(x):= L\tilde{\gamma }(\varphi (x))$. Note that $\gamma _{n}$ is parametrized by arclength and that $\hat{\gamma }=\gamma$ holds. Then, $\gamma _{n} \to \hat{\gamma }=\gamma$ in $C^{1}$ for $n \to \infty$ by (56). Again, by the scaling property of the energies and the invariance under reparametrization we deduce with (56)

$$\begin{aligned} \limsup_{n \to \infty} L^{\frac{n-2}{n}} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\gamma _{n}) \overset{\text{(9)}}{=} \limsup_{n \to \infty} \bigl( \mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}( \tilde{ \gamma }_{n})\leq \mathcal {R}(\tilde{\gamma })=\mathcal {R}(\gamma ). \end{aligned}$$

Hence, it remains to prove the statement of Theorem 1.3 for $L=1$, and for that we take a general sequence $(\gamma _{n})_{n\in \mathbb{N}}\subset C^{1,1}_{\mathrm{ {ia}}}( \mathbb{R}/ \mathbb{Z},\mathbb{R}^{3})$ with $\gamma _{n}\to \gamma$ in $C^{1}$ as $n\to \infty$.

By Lemma 5.2 for $q:=k$

$$\begin{aligned} (\mathrm {TP}_{k})^{\frac{1}{k}}(\gamma )\leq \liminf _{n \to \infty} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\gamma _{n})= \lim_{n\to \infty}\inf _{n \geq k} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\gamma _{n}). \end{aligned}$$

(57)

For $k \leq n$ we apply Lemma 5.3 to $\gamma :=\gamma _{n}$ and $m:=n$ to find

$$\begin{aligned} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\gamma _{n}) \leq \biggl( \frac{4\mathscr{L}(\gamma _{n})^{2} n(n-1)}{n^{2}} \biggr)^{\frac{1}{k}- \frac{1}{n}} \bigl( \mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\gamma _{n}). \end{aligned}$$

Together with (57) and $\mathscr{L}(\gamma _{n})=1$ for all $n\in \mathbb{N}$, this yields

$$\begin{aligned} (\mathrm {TP}_{k})^{\frac{1}{k}}(\gamma ) \leq \lim _{n \to \infty} \inf_{n \geq k} \biggl(\frac{4n(n-1)}{n^{2}} \biggr)^{\frac{1}{k}- \frac{1}{n}} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\gamma _{n}). \end{aligned}$$

(58)

Now, we have

$$\begin{aligned} \lim_{n \to \infty} { \biggl(\frac{4n(n-1)}{n^{2}} \biggr)^{ \frac{1}{k}-\frac{1}{n}}}&= \lim_{n \to \infty} \exp \biggl( \biggl(\frac{1}{k}- \frac{1}{n} \biggr)\log \biggl( \frac{4n(n-1)}{n^{2}} \biggr) \biggr)=\exp \biggl(\frac{1}{k}\log (4) \biggr)=4^{\frac{1}{k}}. \end{aligned}$$

Combining this with the pointwise convergence in Lemma 5.1 and (58) we arrive at the desired liminf inequality:

$$\begin{aligned} \mathcal {R}(\gamma )&\stackrel {}{=}\lim_{k \to \infty} (\mathrm {TP}_{k})^{ \frac{1}{k}}(\gamma ) \stackrel {\text{(58)}}{\leq } \lim _{k \to \infty} \lim_{n \to \infty} \inf_{n \geq k} \biggl(\frac{4n(n-1)}{n^{2}} \biggr)^{\frac{1}{k}-\frac{1}{n}} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}( \gamma _{n}) \\ &\stackrel {}{=}\lim_{k \to \infty} 4^{\frac{1}{k}} \liminf _{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}(\gamma _{n}) \stackrel {}{=}\liminf_{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}( \gamma _{n}). \end{aligned}$$

To verify the limsup inequality let $\gamma \in C_{\mathrm{ {ia}}}^{1,1} ( \mathbb{R}/ \mathbb{Z},\mathbb{R}^{3} )$, and for $n \in \mathbb{N}$ set $s_{n,i}:=\frac{i}{n}$ for $i=0,\dots,n$. Then, we have $|s_{n,i+1}-s_{n,i}|=\frac{1}{n}$ for all $i=0,\dots,n-1$, and therefore a sequence of $(c_{1}-c_{2})$-distributed partitions with $c_{1}=c_{2}=1$; see Definition 2.8. Now, we follow the proof of Theorem 4.4. However, since γ is now a $C^{1,1}$-curve, we do not have to work with convolutions, but can follow the proof for γ directly. By Lemma 2.9 there exists for n sufficiently large a γ-interpolating, proper, and balanced biarc curve $\tilde{\beta}_{n}$ interpolating the point-tangent pairs $(([\gamma (s_{n,i}),\gamma '(s_{n,i})],[\gamma (s_{n,i+1}), \gamma '(s_{n,i+1})]))_{i=0, \dots, n-1}$. Then, we obtain by Theorem 2.11 that $\mathscr{L}(\tilde{\beta}_{n})\to \mathscr{L}(\gamma )=1$ as $n\to \infty$. Let $\varphi _{n}$ be the reparametrization function from [31, Appendix A] and set $\tilde{B}_{n}:=\tilde{\beta}_{n} \circ \varphi _{n}$. Then, by Theorem 2.12, we have $\tilde{B}_{n} \to \gamma$ in $C^{1}$ for $n \to \infty$. Setting $B_{n}:=\mathscr{L}(\tilde{\beta}_{n})^{-1} \tilde{B}_{n}$ we obtain as in the proof of Theorem 4.4 that $B_{n} \to \gamma$ in $C^{1}$ for $n \to \infty$. Let $\beta _{n}$ be the arclength parametrization of $B_{n}$. By Lemma B.3 we finally arrive at $\beta _{n} \to \gamma$ in $C^{1}$ for $n \to \infty$. The biarc curves $\beta _{n}$ are only reparametrized versions of $\tilde{\beta}_{n}$ rescaled by the factor $\mathscr{L}(\tilde{\beta}_{n})^{-1}$, so that we can show exactly as in the proof of Theorem 4.4 that $\beta _{n} \in \mathcal{B}_{n}$ for n sufficiently large. Moreover, due to the $C^{1}$-convergence towards γ, the $\beta _{n}$ are also injective for n large enough. Since $\beta _{n}$ is scaled to unit length and parametrized by arclength, we have $\beta _{n} \in C_{\mathrm{ {ia}}}^{1,1}( \mathbb{R}/ \mathbb{Z},\mathbb{R}^{3})$ for n sufficiently large. Set $L_{n}:=\mathscr{L}(\tilde{\beta}_{n})$. By the scaling property of the discrete tangent-point energy (9) and its parameter invariance we have

$$\begin{aligned} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}( \beta _{n})= L_{n}^{1-\frac{2}{k}} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\tilde{\beta}_{n})\quad \text{for any }k > 2. \end{aligned}$$

(59)

Abbreviating $x_{i,j}:= \frac{2\operatorname{dist}(\gamma (s_{n,i})+\mathbb{R}\gamma '(s_{n,i}),\gamma (s_{n,j}))}{ \vert \gamma (s_{n,i})-\gamma (s_{n,j}) \vert ^{2}}$ for $i,j=0,\dots,n-1$ with $i\neq j$ we can write and estimate for sufficiently large $n\in \mathbb{N}$

$$\begin{aligned} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\beta _{n})&\stackrel {\text{(59)}}{=}L_{n}^{1-\frac{2}{k}} \bigl( \mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\tilde{ \beta}_{n}) \stackrel {}{=}L_{n}^{1-\frac{2}{k}} \Biggl(\sum _{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} x_{i,j}^{k} \lambda _{i} \lambda _{j} \Biggr)^{\frac{1}{k}} \\ & \stackrel {}{=}L_{n}^{1-\frac{2}{k}} \Biggl(\frac{1}{n(n-1)} \sum _{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} n(n-1) x_{i,j}^{k} \lambda _{i} \lambda _{j} \Biggr)^{\frac{1}{k}} \\ & \stackrel {}{\leq }L_{n}^{1-\frac{2}{k}} \biggl(\frac{4L_{n}^{2} n (n-1)}{n^{2}} \biggr)^{\frac{1}{k}} \Biggl(\frac{1}{n(n-1)} \sum _{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} x_{i,j}^{k} \Biggr)^{\frac{1}{k}}. \end{aligned}$$

(60)

Here, we used (7) for $\tilde{\beta}_{n}\in \mathcal{B}_{n}$, which can be verified for n sufficiently large exactly as for the $\tilde{\beta}_{n}^{k}$ in Step 4 of the proof of Theorem 4.4; see in particular (52). Now, observe that by [1, E3.4] applied to the discrete measure $\mu:=\sum_{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} \delta _{( s_{n,i},s_{n,j})}$ with $\mu ((\mathbb{R}/ L\mathbb{Z})^{2})=n(n-1)$ we have

$$\begin{aligned} \lim_{k \to \infty}{ \Biggl( \frac{1}{n(n-1)} \sum _{i=0}^{n-1} \sum _{j=0,j\neq i}^{n-1} x_{i,j}^{k} \Biggr)^{\frac{1}{k}}}=\max_{i,j=0,\dots,n-1,i\neq j} x_{i,j}. \end{aligned}$$

(61)

With $\lim_{k \to \infty} L_{n}^{1-\frac{2}{k}} ( \frac{4L_{n}^{2} n (n-1)}{n^{2}} )^{\frac{1}{k}}=L_{n}$. we obtain by means of (60), (61), and [30, Lemma 2.5] for n sufficiently large

$$\begin{aligned} \begin{aligned} \limsup_{k \to \infty} \bigl( \mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\beta _{n}) & \overset{\text{(60)},\text{(61)}}{\leq} L_{n} \max _{i,j=0,\dots,n-1,i\neq j} x_{i,j}\leq L_{n} \sup _{s,t \in \mathbb{R}/ \mathbb{Z}, s\neq t} \frac{2\operatorname{dist}(\gamma (s)+\mathbb{R}\gamma '(s),\gamma (t))}{ \vert \gamma (s)-\gamma (t) \vert ^{2}} \\ &=L_{n} \sup_{s,t \in \mathbb{R}/ \mathbb{Z}, s\neq t} \frac{1}{r_{\mathrm{ {tp}}} (\gamma (s),\gamma (t))} =L_{n} \mathcal {R}(\gamma ). \end{aligned} \end{aligned}$$

(62)

Now, let $k \geq n$. By virtue of Lemma 5.3 applied to $\beta _{n}$, $m:=k$ and replacing the index k in that lemma by n here, we have $(\mathcal {E}_{n}^{n} )^{\frac{1}{n}}(\beta _{n})\leq (\frac{4L_{n} n (n-1)}{n^{2}} )^{\frac{1}{n}-\frac{1}{k}} ( \mathcal {E}_{k}^{n})^{\frac{1}{k}}(\beta _{n})$, which leads to

$$\begin{aligned} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}( \beta _{n}) &\leq \limsup_{k \to \infty} \biggl( \frac{4L_{n} n (n-1)}{n^{2}} \biggr)^{\frac{1}{n}-\frac{1}{k}} \bigl(\mathcal {E}_{k}^{n} \bigr)^{\frac{1}{k}}(\beta _{n})\stackrel {\text{(62)}}{\leq } \biggl(\frac{4L_{n} n (n-1)}{n^{2}} \biggr)^{\frac{1}{n}} L_{n} \mathcal {R}(\gamma ) \end{aligned}$$

(63)

for n sufficiently large. Finally, taking the limsup yields the desired limsup inequality

$$\begin{aligned} \limsup_{n \to \infty} \bigl(\mathcal {E}_{n}^{n} \bigr)^{\frac{1}{n}}( \beta _{n}) \stackrel {\text{(63)}}{\leq } \limsup_{n \to \infty} \underbrace{ \biggl( \frac{4L_{n} n (n-1)}{n^{2}} \biggr)^{\frac{1}{n}}}_{\to 1} \underbrace{L_{n}}_{\to \mathscr{L}( \gamma )=1} \mathcal {R}(\gamma )=\mathcal {R}(\gamma ). \end{aligned}$$

 □

Proof of Corollary 1.4.

Apply Lemma 4.6 to the metric space $X=C_{\mathrm{ {ia}}}^{1,1} ( \mathbb{R}/ \mathbb{Z}, \mathbb{R}^{3} ) \cap \mathcal{K}$ with the metric induced by the $C^{1}$-norm. Note as in the proof of Corollary 1.2 that according to [25] the knot class $\mathcal{K}$ is stable under $C^{1}$-convergence. Since $\mathcal {R}(\gamma )<\infty$ holds, we obtain by [16, Lemma 2] that $\gamma \in C_{\mathrm{ {ia}}}^{1,1} ( \mathbb{R}/ \mathbb{Z}, \mathbb{R}^{3} )$. □

Appendices

Appendix A: Convergence of convolutions in $W^{2-\frac{1}{q},q}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$

For fixed $L>0$, $s\in (0,1)$ and $\rho \in [1,\infty )$ define the seminorm $[f]_{s,\rho}$ of an L-periodic locally ρ-integrable function $f:\mathbb{R}\to \mathbb{R}^{n}$ as

$$\begin{aligned}{} [f]_{s,\rho}:= \int _{ \mathbb{R}/ L\mathbb{Z}} \int _{ \mathbb{R}/ L\mathbb{Z}} \frac{ \vert f(x)-f(y) \vert ^{\rho}}{ \vert x-y \vert _{\mathbb{R}/L\mathbb{Z}}^{1+s\rho}} \,\mathrm {d}x \,\mathrm {d}y, \end{aligned}$$

(64)

where $|x-y|_{\mathbb{R}/L\mathbb{Z}}$ denotes the periodic distance on $\mathbb{R}$ defined in (13). Then, the periodic fractional⁴Sobolev space $W^{1+s,\rho}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{n})$ consists of those Sobolev functions $f\in W^{1,\rho}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{n})$ whose weak derivatives $f'$ have a finite seminorm $[f']_{s,\rho}$. The norm on $W^{1+s,\rho}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{n})$ is given by $(\|f\|_{W^{1,\rho}}+ [f']_{s,\rho})^{\frac{1}{\rho}}$.

Proof of Theorem 4.2

The case $\rho =\frac{1}{s}$ is treated in [6, Theorem 1.3], so we may assume from now on that $\rho >\frac{1}{s}$.

Step 1: According to Morrey–Sobolev embedding [21, Theorem A.2] we have $\gamma \in C^{1}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$, which implies that $\gamma '$ is of vanishing mean oscillation, in short $\gamma '\in \text{VMO}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$, that is $\lim_{r \to 0} (\sup_{x \in \mathbb{R}/ L\mathbb{Z}} \frac{1}{2r}( \int _{B_{r}(x)} |\gamma '(y) - {\gamma '}_{x,r} |\,\mathrm {d}y ) )=0$, where ${\gamma '}_{x,r}:= \frac{1}{2r}\int _{B_{r}(x)} \gamma '(z)\,\mathrm {d}z$ denotes the integral mean. Indeed, $\gamma '$ is uniformly continuous so that for every $\varepsilon >0$ there exists a $\delta =\delta (\varepsilon )>0$ such that $\vert \gamma '(x)-\gamma '(y) \vert <\frac{\varepsilon}{2}$ for all $x \in \mathbb{R}/ L\mathbb{Z}$ and $y \in B_{\delta}(x)$. Let $0< r<\delta$ and $x \in \mathbb{R}/ L\mathbb{Z}$. Then,

$$\begin{aligned} \frac{1}{2r} \int _{B_{r}(x)} \bigl\vert \gamma '(y) - {\gamma '}_{x,r} \bigr\vert \,\mathrm {d}y & \leq \sup _{y \in B_{r}(x)} \bigl\vert \gamma '(y) - \gamma '(x) \bigr\vert + \bigl\vert \gamma '(x) - {\gamma '}_{x,r} \bigr\vert \\ & \le \frac{\varepsilon}{2} + \frac{1}{2r} \int _{B_{r}(x)} \bigl\vert \gamma '(x)-\gamma '(z) \bigr\vert \,\mathrm {d}z < \varepsilon\quad \text{for any }x\in \mathbb{R}/L \mathbb{Z}, \end{aligned}$$

thus $\sup_{x \in \mathbb{R}/ L\mathbb{Z}} \frac{1}{2r}(\int _{B_{r}(x)} |\gamma '(y) - { \gamma '}_{x,r} |\,\mathrm {d}y ) <\varepsilon$, which implies that $\gamma '\in \text{VMO}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ since $\varepsilon >0$ was arbitrary.

Step 2: For the lengths $L_{\varepsilon}:=\mathscr{L}(\gamma _{\varepsilon})$ and $L:=\mathscr{L}(\gamma )$ we estimate

$$\begin{aligned} \vert L_{\varepsilon}-L \vert &\leq \int _{0}^{L} \bigl\vert \bigl\vert \gamma '_{\varepsilon}(x) \bigr\vert - \bigl\vert \gamma '(x) \bigr\vert \bigr\vert \,\mathrm {d}x \leq \bigl\Vert \bigl\vert \gamma '_{\varepsilon } \bigr\vert - \bigl\vert \gamma ' \bigr\vert \bigr\Vert _{C^{0}} L\stackrel{}{ \underset{\varepsilon \to 0}{\longrightarrow}}0, \end{aligned}$$

(65)

since $\gamma '\in \text{VMO}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$ allows us to apply [6, Theorem 1.1] to deduce that $\vert \gamma '_{\varepsilon } \vert$ converges uniformly to $|\gamma '|=1$ as ε tends to 0. Therefore, there is an $\varepsilon _{0}>0$ such that

$$\begin{aligned} \frac{1}{2}\leq \biggl\vert \frac{L}{L_{\varepsilon}} \gamma _{\varepsilon}' (x) \biggr\vert \leq 2 \quad \text{for any }x\in \mathbb{R}/ L\mathbb{Z}, \varepsilon \in (0,\varepsilon _{0}]. \end{aligned}$$

(66)

Step 3: Since the convolutions $\gamma _{\varepsilon}$ converge to γ in $C^{1}$ we obtain by means of (65) that also the rescalings $L\gamma _{\varepsilon}/L_{\varepsilon}$ converge towards γ in $C^{1}$. According to Lemma B.3 we obtain

$$\begin{aligned} \Vert \tilde{\gamma }_{\varepsilon}-\gamma \Vert _{C^{1}}\longrightarrow 0 \quad\text{{as $\varepsilon \to 0$.}} \end{aligned}$$

(67)

Step 4: It remains to show that $[\tilde{\gamma}'_{\varepsilon }- \gamma ' ]_{s,\rho} \to 0$ holds as $\varepsilon \to 0$, since then, together with (67), we have established $\|\tilde{\gamma }_{\varepsilon}-\gamma \|_{W^{1+s,\rho}}\to 0$ as $\varepsilon \to 0$. Abbreviating the integrand of the seminorm by $I_{\varepsilon}(x,y):= \frac{ \vert (\tilde{\gamma}'_{\varepsilon}(x)- \gamma '(x) )- (\tilde{\gamma}'_{\varepsilon}(y)-\gamma '(y) ) \vert ^{\rho}}{|x-y|_{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}}$ we want to apply Vitali’s theorem (see, e.g., [1, 3.23]) to prove $\|I_{\varepsilon}\|_{L^{1}} \to 0$ as $\varepsilon \to 0$. Since we have a compact domain it suffices to show that the sequence $(I_{\varepsilon} )_{\varepsilon >0}$ is uniformly integrable and converges pointwise to 0 a.e. on $\mathbb{R}/ L\mathbb{Z}\times \mathbb{R}/ L\mathbb{Z}$. The pointwise convergence $I_{\varepsilon}(x,y)\to 0$ (even for all $x\neq y$) follows from the $C^{1}$-convergence (67).

Hence, we need to show the uniform integrability. In the obvious inequality

$$\begin{aligned} I_{\varepsilon}(x,y)\leq 2^{\rho -1} \biggl[ \frac{ \vert \tilde{\gamma}'_{\varepsilon}(x)-\tilde{\gamma}'_{\varepsilon}(y) \vert ^{\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} + \frac{ \vert \gamma '(x)-\gamma '(y) \vert ^{\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \biggr] \end{aligned}$$

(68)

we estimate both summands on the right-hand side separately.

This is easy for the second summand. Fix $\tilde{\varepsilon}>0$. Since γ is in $W^{1+s,\rho}( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3})$, we find a $\delta _{1}=\delta _{1}(\tilde{\varepsilon})>0$ such that for every measurable subset $E \subset (\mathbb{R}/ L\mathbb{Z})^{2}$ with $|E|<\delta _{1}$ we obtain

$$\begin{aligned} \int \int _{E} \frac{ \vert \gamma '(x)-\gamma '(y) \vert ^{\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}}\,\mathrm {d}x\,\mathrm {d}y < \frac{\tilde{\varepsilon}}{2^{\rho}}. \end{aligned}$$

(69)

Regarding the first summand in (68) we consider the arclength function $s_{\varepsilon}(x):= \int _{0}^{x} | \frac{L}{L_{\varepsilon}}\gamma '_{\varepsilon}(z) |\,\mathrm {d}z$ such that $s_{\varepsilon}(0)=0$. From (66) the derivative $s'_{\varepsilon}(z)=|\frac{L}{L_{\varepsilon}}\gamma '_{\varepsilon}(z) |$ is uniformly bounded away from 0 for all $\varepsilon \in (0,\varepsilon _{0}]$. As a consequence, $s_{\varepsilon}$ is for $\varepsilon \in (0,\varepsilon _{0}]$ invertible. Let $\tilde{s}_{\varepsilon}$ denote the inverse function of $s_{\varepsilon}$. As a next step, we will show $|s_{\varepsilon}(x)-s_{\varepsilon}(y)|_{ \mathbb{R}/ L\mathbb{Z}}\geq \frac{1}{2}|x-y|_{ \mathbb{R}/ L\mathbb{Z}}$ for $x,y \in \mathbb{R}/ L\mathbb{Z}$ and $\varepsilon \in (0,\varepsilon _{0}]$. Let $0\leq x < y < L$, so that by monotonicity $0\leq s_{\varepsilon}(x)\leq s_{\varepsilon}(y)< L$. First, assume that $|s_{\varepsilon}(x)-s_{\varepsilon}(y)|_{ \mathbb{R}/ L\mathbb{Z}}=|s_{\varepsilon}(x)-s_{ \varepsilon}(y)|=s_{\varepsilon}(y)-s_{\varepsilon}(x)$. Then, we estimate by means of (66)

$$\begin{aligned} \bigl\vert s_{\varepsilon}(x)-s_{\varepsilon}(y) \bigr\vert _{ \mathbb{R}/ L\mathbb{Z}} = \int _{x}^{y} \biggl\vert \frac{L}{L_{\varepsilon}} \gamma '_{\varepsilon}(z) \biggr\vert \,\mathrm {d}z \overset{\text{ (66)}}{\geq} \frac{1}{2}(y-x)\geq \frac{1}{2} \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$

If $|s_{\varepsilon}(x)-s_{\varepsilon}(y)|_{ \mathbb{R}/ L\mathbb{Z}}=L-(s_{\varepsilon}(y)-s_{ \varepsilon}(x))$, then again by (66)

$$\begin{aligned} \bigl\vert s_{\varepsilon}(x)-s_{\varepsilon}(y) \bigr\vert _{ \mathbb{R}/ L\mathbb{Z}}&= L- \int _{x}^{y} \biggl\vert \frac{L}{L_{\varepsilon}} \gamma '_{\varepsilon}(z) \biggr\vert \,\mathrm {d}z= \int _{0}^{x} \biggl\vert \frac{L}{L_{\varepsilon}} \gamma '_{\varepsilon}(z) \biggr\vert \,\mathrm {d}z + \int _{y}^{L} \biggl\vert \frac{L}{L_{\varepsilon}} \gamma '_{ \varepsilon}(z) \biggr\vert \,\mathrm {d}z \\ &\overset{\text{(66)}}{\geq} \frac{1}{2} \bigl(L-(y-x) \bigr) \geq \frac{1}{2} \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$

In particular, this yields for the inverse function

$$\begin{aligned} \bigl\vert \tilde{s}_{\varepsilon}(x)-\tilde{s}_{\varepsilon}(y) \bigr\vert _{ \mathbb{R}/ L\mathbb{Z}}\leq 2 \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}} \quad \text{for any }x, y \in \mathbb{R}/ L\mathbb{Z}, \varepsilon \in (0,\varepsilon _{0}]. \end{aligned}$$

(70)

Due to (65) there exists a constant $c>0$ such that

$$\begin{aligned} 2^{1+(1+s)\rho} \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert ^{\rho} \biggl[1+2^{\rho} \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert ^{\rho } \biggr] \leq c \quad \text{for any }\varepsilon >0. \end{aligned}$$

(71)

Now, we estimate pointwise for $x \neq y$ with Jensen’s inequality and by (70)

$$\begin{aligned} J_{\varepsilon}(x,y)&:= \frac{ \vert \tilde{\gamma}'_{\varepsilon}(x)- \tilde{\gamma}'_{\varepsilon}(y) \vert ^{\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} = \frac{ \vert \frac{L}{L_{\varepsilon}} \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(x))\tilde{s}_{\varepsilon}'(x) -\frac{L}{L_{\varepsilon}} \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \tilde{s}_{\varepsilon}'(y) \vert ^{\rho}}{ \vert \tilde{s}_{\varepsilon}(x)-\tilde{s}_{\varepsilon}(y) \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \frac{ \vert \tilde{s}_{\varepsilon}(x)-\tilde{s}_{\varepsilon}(y) \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \\ &\overset{\text{(70)}}{ \leq} 2^{1+s\rho} \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert ^{\rho } \biggl[ \frac{ \vert \tilde{s}_{\varepsilon}'(x) \vert ^{\rho } \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(x)) -\gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \vert ^{\rho}}{ \vert \tilde{s}_{\varepsilon}(x) -\tilde{s}_{\varepsilon}(y) \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} + \frac{ \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \vert ^{\rho } \vert \tilde{s}_{\varepsilon}'(x)-\tilde{s}_{\varepsilon}'(y) \vert ^{\rho}}{ \vert \tilde{s}_{\varepsilon}(x)-\tilde{s}_{\varepsilon}(y) \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \biggr]. \end{aligned}$$

Together with $|\gamma _{\varepsilon}'|\le 1$, $|s_{\varepsilon}'(x)|= |\frac{L}{L_{\varepsilon}} \gamma _{ \varepsilon}'(x)|\in [\frac {1}{2},2 ]$ due to (66), $|\tilde{s}_{\varepsilon}'(x)|= {|\frac{L}{L_{\varepsilon}} \gamma _{ \varepsilon}'(\tilde{s}_{\varepsilon}(x))|^{-1}} \in [ \frac {1}{2},2 ]$ for $\varepsilon \in (0,\varepsilon _{0}]$, and the estimate

$$\begin{aligned} \bigl\vert \tilde{s}_{\varepsilon}'(x)-\tilde{s}_{\varepsilon}'(y) \bigr\vert &\stackrel {}{=}\biggl\vert { \biggl\vert \frac{L}{L_{\varepsilon}} \gamma _{\varepsilon}' \bigl(\tilde{s}_{\varepsilon}(x) \bigr) \biggr\vert ^{-1}}-{ \biggl\vert \frac{L}{L_{\varepsilon}} \gamma _{\varepsilon}' \bigl(\tilde{s}_{ \varepsilon}(y) \bigr) \biggr\vert ^{-1}} \biggr\vert \\ &\stackrel {}{\leq }\biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert \frac{ \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(x))-\gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \vert }{ \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(x)) \vert \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \vert } \\ &\stackrel {\text{(66)}}{\leq } 4 \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert \bigl\vert \gamma _{\varepsilon}' \bigl( \tilde{s}_{\varepsilon}(x) \bigr)-\gamma _{\varepsilon}' \bigl(\tilde{s}_{ \varepsilon}(y) \bigr) \bigr\vert , \end{aligned}$$

we obtain for all $\varepsilon \in (0,\varepsilon _{0}]$ the inequality

$$\begin{aligned} J_{\varepsilon}(x,y)\leq 2^{1+s\rho} \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert ^{\rho} \biggl[2^{\rho}+4^{\rho} \biggl\vert \frac{L}{L_{\varepsilon}} \biggr\vert ^{\rho } \biggr] \frac{ \vert \gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(x))-\gamma _{\varepsilon}'(\tilde{s}_{\varepsilon}(y)) \vert ^{\rho}}{ \vert \tilde{s}_{\varepsilon}(x)-\tilde{s}_{\varepsilon}(y) \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \stackrel {\text{(71)}}{\leq }c A_{\varepsilon} \bigl(\psi _{\varepsilon}(x,y) \bigr) \end{aligned}$$

(72)

for $A_{\varepsilon}(x,y):= \frac{ \vert \gamma _{\varepsilon}'(x) -\gamma _{\varepsilon}'(y) \vert ^{\rho}}{|x-y|_{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}}$ and the transformation $\psi _{\varepsilon }: (\mathbb{R}/ L\mathbb{Z})^{2} \to (\mathbb{R}/ L\mathbb{Z})^{2}$ sending $(x,y)$ to $(\tilde{s}_{\varepsilon}(x),\tilde{s}_{\varepsilon}(y))$. Observe that by (66), $\psi _{\varepsilon}$ is bi-Lipschitz, since $\operatorname{det}(D\psi _{\varepsilon}(x,y))= |\tilde{s}_{\varepsilon}'(x)|| \tilde{s}_{\varepsilon}'(y)| \in [\frac {1}{4},4]$. This implies by (72) that

$$\begin{aligned} J_{\varepsilon}(x,y)\leq 4c A_{\varepsilon} \bigl(\psi _{\varepsilon}(x,y) \bigr) \bigl\vert \operatorname{det} \bigl(D\psi _{\varepsilon}(x,y) \bigr) \bigr\vert \quad \text{for any }\varepsilon \in (0,\varepsilon _{0}]. \end{aligned}$$

Let now $E \subset (\mathbb{R}/ L\mathbb{Z})^{2}$. By a change of variables

$$\begin{aligned} \int \int _{E} J_{\varepsilon}(x,y) \,\mathrm {d}x \,\mathrm {d}y&\leq 4c \int \int _{E} A_{\varepsilon} \bigl(\psi _{\varepsilon}(x,y) \bigr) \bigl\vert \operatorname{det} \bigl(D\psi _{\varepsilon}(x,y) \bigr) \bigr\vert \,\mathrm {d}x \,\mathrm {d}y \\ & =4c \int \int _{\psi _{\varepsilon}(E)} A_{\varepsilon}(x,y) \,\mathrm {d}x \,\mathrm {d}y\quad \text{for any }\varepsilon \in (0,\varepsilon _{0}]. \end{aligned}$$

(73)

It is well known that the standard convolution $\gamma _{\varepsilon}$ converges in $W^{1+s, \rho}$ to γ; see, e.g., [13, Lemma 11], which according to Vitali’s theorem implies that the $A_{\varepsilon}(x,y)$ are uniformly integrable. In particular, for given $\tilde{\varepsilon}>0$ there exists $\delta _{2}=\delta _{2}(\tilde{\varepsilon})>0$ such that if $|\psi _{\varepsilon}(E)|<\delta _{2}$, we have

$$\begin{aligned} \int \int _{\psi _{\varepsilon}(E)} A_{\varepsilon}(x,y) \,\mathrm {d}x \,\mathrm {d}y < \frac{\tilde{\varepsilon}}{c 2^{\rho +2}} \quad \text{for any }\varepsilon >0. \end{aligned}$$

(74)

Since $\psi _{\varepsilon}$ is uniformly Lipschitz continuous for $\varepsilon \in (0,\varepsilon _{0}]$, there exists a $\delta _{3}>0$ such that $|E| <\delta _{3}$ implies $|\psi _{\varepsilon}(E) |<\delta _{2}$. Now, set $\tilde{\delta}:=\min \{ \delta _{1}, \delta _{3}\}$ so that for any set $E\subset ( \mathbb{R}/ L\mathbb{Z})^{2}$ with $|E|<\tilde{\delta}$ we infer by means of (68), (73), (69), and (74) that

$$\begin{aligned} \int \int _{E} I_{\varepsilon}(x,y)\,\mathrm {d}x \,\mathrm {d}y& \stackrel { \text{(68)},\text{(73)}}{\leq } 2^{\rho -1} \biggl[ 4c \int \int _{\psi _{\varepsilon}(E)} A_{ \varepsilon}(x,y) \,\mathrm {d}x \,\mathrm {d}y+ \int \int _{E} \frac{ \vert \gamma '(x)-\gamma '(y) \vert ^{\rho}}{ \vert x-y \vert _{ \mathbb{R}/ L\mathbb{Z}}^{1+s\rho}} \,\mathrm {d}x \,\mathrm {d}y \biggr] \\ &\stackrel {\text{(69)},\text{ (74)}}{< } 2^{\rho -1} \biggl[4c \frac{\tilde{\varepsilon}}{c 2^{\rho +2}} + \frac{\tilde{\varepsilon}}{2^{\rho}} \biggr] \stackrel {}{=}\tilde{ \varepsilon} \quad \text{for any }\varepsilon \in (0,\varepsilon _{0}]. \end{aligned}$$

Hence, $(I_{\varepsilon} )_{\varepsilon \in (0,\varepsilon _{0}]}$ is uniformly integrable. □

Appendix B: Quantitative analysis of $C^{1}$-curves

Lemma B.1

(Injective $C^{1}$-curves are bi-Lipschitz)

For any curve $\gamma \in C^{1}_{\mathrm{ {ia}}}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})$ there is a constant $c_{\gamma }>0$ such that

$$\begin{aligned} \vert t-s \vert _{\mathbb{R}/L\mathbb{Z}}\le c_{\gamma } \bigl\vert \gamma (t)-\gamma (s) \bigr\vert \quad \textit{for any }t,s\in \mathbb{R}. \end{aligned}$$

(75)

Proof

Using the Taylor expansion

$$\begin{aligned} \gamma (s+h)-\gamma (s)= \int _{s}^{s+h}\gamma '(\tau ) \,\mathrm {d}\tau = \gamma '(s)h+ \int _{s}^{s+h} \bigl(\gamma '(\tau )- \gamma '(s) \bigr) \,\mathrm {d}\tau \end{aligned}$$

(76)

we choose $h_{0}=h_{0}(\gamma )\in (0,\frac{L}{2}]$ such that $\omega _{\gamma '}(h_{0})\le \frac{1}{2}$ to infer for all $s\in \mathbb{R}$

$$\begin{aligned} \bigl\vert \gamma (s+h)-\gamma (s) \bigr\vert \ge \bigl(1- \omega _{\gamma '}(h_{0}) \bigr) \vert h \vert \ge \frac{1}{2} \vert h \vert = \frac{1}{2} \vert h \vert _{\mathbb{R}/L\mathbb{Z}} \quad \text{for any }\vert h \vert \le h_{0}. \end{aligned}$$

(77)

On the other hand, since γ is injective, we find a constant $\delta _{0}=\delta _{0}(\gamma )>0$ such that

$$\begin{aligned} \bigl\vert \gamma (s+h)-\gamma (s) \bigr\vert \ge \delta _{0}\ge \frac{2\delta _{0}}{L} \vert h \vert =\frac{2\delta _{0}}{L} \vert h \vert _{\mathbb{R}/L \mathbb{Z}} \quad \text{for any }s\in \mathbb{R}, h_{0}\le \vert h \vert \le \frac{L}{2}, \end{aligned}$$

(78)

which implies (75) for $c_{\gamma }:=\max \{2, \frac{L}{2\delta _{0}}\}$. □

Lemma B.2

Let $\gamma \in C^{1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )$ and $h\in (0,\frac{L}{2}]$ such that $\omega _{\gamma '}(h)<1$ and $h\leq \frac{L}{2}$. Then, for every $s \in \mathbb{R}$,

$$\begin{aligned} &1-\omega _{\gamma '}(h)\leq \frac{ \langle \gamma (s+h)-\gamma (s),\gamma '(s) \rangle}{ \vert \gamma (s+h )-\gamma (s ) \vert } \leq 1 , \end{aligned}$$

(79)

$$\begin{aligned} &1-\omega _{\gamma '}(h)\leq \frac{ \langle \gamma (s+h)-\gamma (s),\gamma '(s+h) \rangle}{ \vert \gamma (s+h )-\gamma (s ) \vert } \leq 1. \end{aligned}$$

(80)

Proof

Since $0< h\leq \frac{L}{2}$ we have $|s+h-s|_{ \mathbb{R}/ L\mathbb{Z}} = h$; see (13). We use the Taylor expansion (76) to estimate

$$\begin{aligned} \bigl\langle \gamma (s+h)-\gamma (s),\gamma '(s) \bigr\rangle \geq h \Bigl[1-\sup_{\tau \in [s,s+h]} \bigl\vert \gamma '(\tau )-\gamma '(s) \bigr\vert \Bigr] \geq h \bigl[1- \omega _{\gamma '}(h) \bigr] \end{aligned}$$

(81)

and analogously

$$\begin{aligned} \bigl\langle \gamma (s+h)-\gamma (s),\gamma '(s+h) \bigr\rangle &\geq h \bigl[1- \omega _{\gamma '}(h) \bigr]. \end{aligned}$$

(82)

Using the above estimate for the inner product and the Lipschitz estimate $|\gamma (s+h)-\gamma (s)|\le h$ we can deduce

$$\begin{aligned} \frac{ \langle \gamma (s+h)-\gamma (s),\gamma '(s) \rangle}{ \vert \gamma (s+h )-\gamma (s ) \vert } \stackrel {\text{(81)}}{\geq } \frac{h(1-\omega _{\gamma '}(h))}{ \vert \gamma (s+h)-\gamma (s) \vert }\geq \frac{h(1-\omega _{\gamma '}(h))}{h}= 1-\omega _{\gamma '}(h). \end{aligned}$$

Applying the Cauchy–Schwarz inequality yields the right part of inequality (79). Thus, statement (79) is shown. In the same manner we can conclude the statement (80) with the Cauchy–Schwarz inequality and estimate (82). □

Lemma B.3

Let $\gamma \in C^{1} ([a,b],\mathbb{R}^{3} )$ satisfy $|\gamma '|\geq v_{\gamma }>0$ and $\mathscr{L}(\gamma )>0$ and let $\Gamma \in C^{1} ([0,\mathscr{L}(\gamma )],\mathbb{R}^{3} )$ be the arclength parametrization. Suppose that $\beta \in C^{1} ([a,b],\mathbb{R}^{3} )$ has equal length, i.e. $\mathscr{L}(\gamma )=\mathscr{L}(\beta )$, and satisfies

$$\begin{aligned} \Vert \gamma - \beta \Vert _{C^{1} ([a,b],\mathbb{R}^{3} )} < \varepsilon \leq \frac{v_{\gamma}}{2}. \end{aligned}$$

(83)

Then, β possesses an arclength parametrization $B\in C^{1} ([0,\mathscr{L}(\gamma )],\mathbb{R}^{3} )$ with

$$\begin{aligned} \Vert \Gamma - B \Vert _{C^{1} ([0,\mathscr{L}(\gamma )],\mathbb{R}^{3} )} \leq \frac{2}{v_{\gamma}}\omega _{\gamma '} \biggl(\frac{(b-a)\varepsilon}{v_{\gamma}} \biggr) + \omega _{\gamma} \biggl(\frac{(b-a)\varepsilon}{v_{\gamma}} \biggr) + \varepsilon \biggl(1+ \frac{2}{v_{\gamma}} \biggr), \end{aligned}$$

(84)

where $\omega _{\gamma}$ denotes the modulus of continuity of γ and $\omega _{\gamma '}$ denotes the modulus of continuity of the tangent $\gamma '$ of γ.

Proof

Without loss of generality, we can assume $\Gamma (s)=\gamma (t(s))$ for $s \in [0,\mathscr{L}(\gamma )]$, where $t: [0,\mathscr{L}(\gamma )] \to [a,b]$ is the inverse function of the arclength function $s(t):=\int _{a}^{t} |\gamma '(u)| \,\mathrm {d}u$ for $t \in [a,b]$. Furthermore, the conditions $|\gamma '|\geq v_{\gamma}$ and (83) imply $|\beta '(t)|\geq \frac{v_{\gamma}}{2}>0$ for all $t \in [a,b]$. Hence, the arclength function of β, $\sigma (t):=\int _{a}^{t} |\beta '(u)|\,\mathrm {d}u$ is therefore also invertible. Let $\tau: [0,\mathscr{L}(\gamma )] \to [a,b]$ be the inverse function of σ and define the arclength parametrization of β as $B(s):=\beta (\tau (s))$ for $s\in [0,\mathscr{L}(\gamma )]$. Now, fix an $s \in [0,\mathscr{L}(\gamma )]$. Then, there exist unique $t,\tau \in [a,b]$ such that $s=s(t)=\sigma (\tau )$. This leads to

$$\begin{aligned} 0 &= \sigma (\tau ) - s(t) = \int _{a}^{\tau} \bigl( \bigl\vert \beta '(u) \bigr\vert - \bigl\vert \gamma '(u) \bigr\vert \bigr) \,\mathrm {d}u - \int _{\tau}^{t} \bigl\vert \gamma '(u) \bigr\vert \,\mathrm {d}u. \end{aligned}$$

(85)

Thus, we can estimate

$$\begin{aligned} v_{\gamma } \vert t-\tau \vert \stackrel {}{\leq }\biggl| \int _{\tau}^{t} \bigl\vert \gamma '(u) \bigr\vert \,\mathrm {d}u \biggr| \stackrel {\text{(85)}}{=} \biggl| \int _{a}^{\tau} \bigl( \bigl\vert \beta '(u) \bigr\vert - \bigl\vert \gamma '(u) \bigr\vert \bigr)\,\mathrm {d}u \biggr| \stackrel {\text{(83)}}{\leq } \varepsilon (b-a). \end{aligned}$$

(86)

Now, we can use (83) and (86) to estimate the distance between Γ and B by

$$\begin{aligned} \bigl\vert \Gamma (s)-B(s) \bigr\vert &\stackrel {}{=}\bigl\vert \gamma (t)-\beta (\tau ) \bigr\vert \stackrel {}{\leq }\bigl\vert \gamma (t)-\gamma (\tau ) \bigr\vert + \bigl\vert \gamma (\tau )-\beta (\tau ) \bigr\vert \\ &\stackrel {\text{(83)}}{\leq } \omega _{\gamma} \bigl( \vert t-\tau \vert \bigr) + \varepsilon \stackrel {\text{(86)}}{\leq } \omega _{\gamma} \biggl(\frac{(b-a)\varepsilon}{v_{\gamma}} \biggr) + \varepsilon. \end{aligned}$$

(87)

With $\tau '(s)=\frac{1}{\sigma '(\tau (s))}= \frac{1}{ \vert \beta '(\tau (s)) \vert }$ we obtain $B'(s)=\beta '(\tau (s))\tau '(s)= \frac{\beta '(\tau (s))}{ \vert \beta '(\tau (s)) \vert }$ and analogously $\Gamma '(s)=\frac{\gamma '(t(s))}{ \vert \gamma '(t(s)) \vert }$. This leads to

$$\begin{aligned} & \bigl\vert \Gamma '(s)-B'(s) \bigr\vert \\ &\quad \stackrel {}{=}\biggl\vert \frac{\gamma '(t)}{ \vert \gamma '(t) \vert } - \frac{\beta '(\tau )}{ \vert \beta '(\tau ) \vert } \biggr\vert \\ &\quad \stackrel {}{\leq }\frac{ \vert \gamma '(t)-\gamma '(\tau ) \vert }{ \vert \gamma '(t) \vert } + \bigl\vert \gamma '(\tau ) \bigr\vert \biggl\vert \frac{1}{ \vert \gamma '(t) \vert } - \frac{1}{ \vert \gamma '(\tau ) \vert } \biggr\vert \\ &\qquad{}+ \frac{ \vert \gamma '(\tau ) - \beta '(\tau ) \vert }{ \vert \gamma '(\tau ) \vert }+ \bigl\vert \beta '(\tau ) \bigr\vert \biggl\vert \frac{1}{ \vert \gamma '(\tau ) \vert } - \frac{1}{ \vert \beta '(\tau ) \vert } \biggr\vert \\ &\quad \stackrel {}{\leq }\frac{1}{v_{\gamma}}\omega _{\gamma '} \bigl( \vert t-\tau \vert \bigr) + \frac{1}{v_{\gamma}}\omega _{\gamma '} \bigl( \vert t-\tau \vert \bigr) + \frac{\varepsilon}{v_{\gamma}} + \frac{\varepsilon}{v_{\gamma}} \stackrel {\text{ (86)}}{\leq } \frac{2}{v_{\gamma}} \biggl( \omega _{\gamma '} \biggl(\frac{(b-a)\varepsilon}{v_{\gamma}} \biggr) + \varepsilon \biggr). \end{aligned}$$

(88)

With (87) and (88), we deduce (84). □