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The Willmore flow with prescribed isoperimetric ratio

Fabian Rupp

Abstract

We introduce a non-local $L^2$-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and convergence to a Helfrich immersion. This is in sharp contrast to the locally constrained flow, where finite time singularities occur.

The Willmore flow with prescribed isoperimetric ratio

Abstract

We introduce a non-local -gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and convergence to a Helfrich immersion. This is in sharp contrast to the locally constrained flow, where finite time singularities occur.

Paper Structure

This paper contains 14 sections, 34 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Geometric and analytic background
  4. Evolution of geometric quantities
  5. Localized energy estimates
  6. Controlling the Lagrange multiplier
  7. The blow-up and its properties
  8. A lower bound on the existence time
  9. Existence of a blow-up
  10. The constrained Lojasiewicz--Simon inequality
  11. Convergence for spheres
  12. An upper bound for beta0
  13. Proof of Lemma 7.3
  14. Higher order evolution

Key Result

Theorem 1

Let $f_0\colon \mathbb{S}^2\to\mathbb{R}^3$ be a smooth immersion with $\mathcal{I}(f_0)=\sigma\in (0,1)$ and such that $\mathcal{W}(f_0)\leq \min\left\{\frac{4\pi}{\sigma},8\pi\right\}$. Then there exists a unique $\sigma$-isoperimetric Willmore flow with initial datum $f_0$. This flow exists for a

Figures (1)

  • Figure 1: The function $F(a)$ in \ref{['lem:F(a)<0']}.

Theorems & Definitions (67)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Proposition 2.1
  • Lemma 2: RuppVolumePreserving
  • Proposition 2.2
  • proof
  • Lemma 3
  • ...and 57 more