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A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies

Elias Döhrer, Alexander Dohmen

Abstract

In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.

A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies

Abstract

In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.

Paper Structure

This paper contains 12 sections, 36 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. An approximate Fenchel-type Theorem
  5. Minimizing functionals depending on the direction of the secant
  6. Minimizing most tangent-point energies
  7. The geometric case $p=2q$
  8. The case $p>2q$
  9. The lower limit case $p=q+1$
  10. The case $p<2q$
  11. Minimizing Fractional Willmore energies
  12. Appendix

Key Result

Theorem 1.1

Let $q\geq 1$, $p\in \intervalco{q+1,2q+1}\cap \intervalcc{2q-2,4q-2}$ or $p=q+1$ and $L>0$. For every $\gamma\in W^{1,1}_{\mathrm{i,r}}({\mathbb{T}}, {\mathbb{R}}^n)$ with $\mathcal{L}(\gamma)=L$, the following holds true. where $\Gamma$ denotes the Gamma function. Additionally, for $q\geq 1, p\in \intervaloo{2q,2q+1}$, the inequality holds among all convex, Lipschitz curves. Equality in0.153mu0

Figures (1)

  • Figure 1: Figure 1 (a) illustrates the self-repulsive regime of $\mathrm{TP}^{(p,q)}$. It consists of the scale-invariant critical case$p=q+2, q>1$ and the subcritical case$p\in (q+2,2q+1), q>1$ (yellow region). For $p\geq 2q+1$, the energy of any closed $C^1_{\mathrm{i,r}}$-curve is infinite. We introduce the line $p=q+1, q\geq 1$, which will be referred to as lower limit case. Even though $\mathrm{TP}^{(p,q)}$ is not a knot energy for $p\in \intervaloo{q+1,q+2}, q\geq 1$, the energy still penalizes certain types of self-intersections. Hence, we refer to this as mild repulsion; see \ref{['section: The lower limit case p=q+1']} for more details. Figure 1 (b) indicates regions where $\mathrm{TP}^{(p,q)}$ is minimized by circles (blue and red region). Note that for $p<q+1$, there is no global minimizer within the class of closed, injective, $W^{1,1}$-curves with fixed length, since the infimum $0$ is only attained for straight lines.

Theorems & Definitions (68)

  • Theorem 1.1: Sharp lower bound for $\mathrm{TP}^{(p,q)}$
  • Theorem 1.2: Uniqueness of minimizers
  • Theorem 1.3
  • Theorem 1.4: Fenchel Theorem for Gauss Maps
  • Theorem 2.1: Abrams, Cantarella, Fu, Ghomi, Howard,0.153mu0.153muABRAMS2003381
  • Theorem 2.2: Theorem $5$,0.153mu0.153muABRAMS2003381
  • Lemma 2.3
  • Proof 1
  • Lemma 2.4
  • Proof 2
  • ...and 58 more