Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Willmore Flow of Complete Surfaces

Long-Sin Li

Abstract

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result by Kuwert and Schätzle in 2002 and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.

Willmore Flow of Complete Surfaces

Abstract

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result by Kuwert and Schätzle in 2002 and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.
Paper Structure (11 sections, 46 theorems, 253 equations)

This paper contains 11 sections, 46 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. General inequalities
  4. Interpolation inequalities
  5. Sobolev inequalities
  6. Geometric inequalities
  7. Evolution equations
  8. Energy estimates
  9. A priori estimates and existence time
  10. Uniqueness
  11. Gap phenomena

Key Result

Lemma 2.1

Given any $x_1,x_2\in\mathbb{R}^n$, $R_1,R_2>0$, and $0<K_1,K_2\leq K$, we can let so that they satisfy (cutoff).

Theorems & Definitions (83)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1: Local convexity implies global convexity
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 73 more