Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved long time existence for the Willmore flow of surfaces of revolution with Dirichlet data

Sascha Eichmann

Abstract

To avoid possible singularities in the Willmore flow, one usually works under an energy threshold provided by the Li-Yau inequality. Here we improve this threshold by also considering parts outside of a possible singularity together with Dirichlet boundary data. We work in the class of surfaces of revolution.

Improved long time existence for the Willmore flow of surfaces of revolution with Dirichlet data

Abstract

To avoid possible singularities in the Willmore flow, one usually works under an energy threshold provided by the Li-Yau inequality. Here we improve this threshold by also considering parts outside of a possible singularity together with Dirichlet boundary data. We work in the class of surfaces of revolution.
Paper Structure (8 sections, 12 theorems, 132 equations, 7 figures)

This paper contains 8 sections, 12 theorems, 132 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Geometric background
  3. Asymptotic geodesic elastica
  4. Minimising with a singular boundary value
  5. Variational Background
  6. Existence with one sided singular boundary
  7. Proof of main Theorem \ref{['1_1']}
  8. Helpful results

Key Result

Theorem 1.1

Let $\gamma_0:[0,1]\rightarrow\mathbb{H}$ be a regular smooth curve satisfying the boundary data eq:1_5, such that Then there is a global, rotational symmetric solution $f:[0,\infty)\times\Sigma\rightarrow \mathbb{R}^3$ to eq:1_4, with initial data $\gamma_0$ and satisfying the boundary data eq:1_5. Moreover $f(t,\cdot)$ converges up to reparametrization smoothly to a Willmore immersion $f_\infty

Figures (7)

  • Figure 1: Boundary data and added sphere caps.
  • Figure 2: Sketch of $c^x$ consisting of one Moebius transformed catenoid and one sphere.
  • Figure 3: Plot of $x\mapsto W^e_{closed}(S({c^x}))$ für $\alpha_-=1$ and $\alpha_+=2$.
  • Figure 4: Plot of $\alpha_+\mapsto \inf_x W^e_{closed}(S({c^x}))$ for $\alpha_-=1$.
  • Figure 5: Plot of $\alpha_+\mapsto \inf_x W^e_{closed}(S({c^x}))$ for $\alpha_-=10$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1: cf. EichmannDiplom Thm. 6.7, cf. MuellerSpenerElasticFlow Lemma 2.9
  • Lemma 3.2
  • proof
  • Definition 4.1: see Def. 3 in ChoksiVeroni
  • Lemma 4.2: see Lemma 3 in ChoksiVeroni
  • Lemma 4.3: see Lemma 4 in ChoksiVeroni
  • Definition 4.4: see Def. 4 and Def. 6 in ChoksiVeroni
  • Theorem 4.5: see Prop. 1 in ChoksiVeroni
  • ...and 8 more