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The Willmore Flow of Tori of Revolution

Anna Dall'Acqua, Marius Müller, Reiner Schätzle, Adrian Spener

Abstract

We study long-time existence and asymptotic behavior for the $L^2$-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below $8π$ then the solution of the Willmore flow converges for $t \rightarrow \infty$ to the Clifford Torus, possibly rescaled and translated. The energy threshold of $8π$ turns out to be optimal for such a convergence result. We give an application to the conformally constrained Willmore minimization problem.

The Willmore Flow of Tori of Revolution

Abstract

We study long-time existence and asymptotic behavior for the -gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below then the solution of the Willmore flow converges for to the Clifford Torus, possibly rescaled and translated. The energy threshold of turns out to be optimal for such a convergence result. We give an application to the conformally constrained Willmore minimization problem.

Paper Structure

This paper contains 17 sections, 33 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. Notation
  4. Curves in the hyperbolic plane
  5. Tori of revolution in $\mathbb{R}^3$
  6. The Willmore flow of tori of revolution
  7. Singularities of the Willmore flow
  8. Dimension reduction
  9. Symmetry of the limit immersion
  10. Rotational symmetry and concentration
  11. Proof of Theorems \ref{['thm:boundhypmain']} and \ref{['thm:MainThmPrecisely']}
  12. Optimality
  13. An application: Energy minimization among conformal constraints
  14. Consistency between extrinsic and intrinsic view
  15. Tensor calculus
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $f: [0,T) \times \mathbb{S}^1 \times \mathbb{S}^1 \rightarrow \mathbb{R}^3$ be a maximal evolution by Willmore flow such that $f(0)$ is a torus of revolution. Then $f(t)$ is a torus of revolution for all $t \in [0,T)$. Suppose that $(\gamma(t))_{t \in [0,T)}$ is a collection of profile curves of then $T = \infty$ and the Willmore flow converges (up to reparametrizations) in $C^k$ for all $k$ t

Theorems & Definitions (75)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: AdrianMarius
  • Lemma 2.3
  • proof
  • Proposition 2.4: A gap theorem for Willmore tori of revolution
  • proof
  • ...and 65 more