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Weighted $\infty$-Willmore Spheres

Ed Gallagher, Roger Moser

Abstract

On the two-sphere $Σ$, we consider the problem of minimising among suitable immersions $f \,\colon Σ\rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $ξ$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H \in \{ \pm \vert \vert ξH \vert \vert_{L^\infty} \}$ away from the nodal set of the PDE system, and $H = 0$ on the nodal set (if it is non-empty).

Weighted $\infty$-Willmore Spheres

Abstract

On the two-sphere , we consider the problem of minimising among suitable immersions the weighted norm of the mean curvature , with weighting given by a prescribed ambient function , subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as of the Euler-Lagrange equations for the approximating problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: away from the nodal set of the PDE system, and on the nodal set (if it is non-empty).
Paper Structure (11 sections, 25 theorems, 116 equations)

This paper contains 11 sections, 25 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. The Penalised $L^p$ Problem
  3. The Penalisation Term
  4. Minimisers of the $L^p$ functional $\mathcal{M}_p^{\sigma} [ \,\mathbin{\vcenter{\hbox{$\m@th\bullet$}{}}}\, ; f_0 ]$
  5. Approximation of $\mathcal{M}_p^{\sigma} [ \,\mathbin{\vcenter{\hbox{$\m@th\bullet$}{}}}\, ; f_0 ]$ Minimisers
  6. Convergence of Approximations
  7. Convergence of Immersions
  8. Uniform Bounds; Notation & Terminology
  9. Convergence of $w$ as $\varepsilon \rightarrow 0$ for fixed $p$
  10. Convergence of Euler-Lagrange as $p \rightarrow \infty$
  11. Behaviour and Existence of $\infty$-Willmore spheres

Key Result

Theorem 2

Let $\xi$ be a given weight function satisfying cond:alph1$\,$--$\,$cond:alph3 and let $f \in \mathcal{G}$ be an $\infty$-Willmore sphere. Assume that the low-energy inequality $\lVert\xi H\rVert_{L^\infty} A^{\frac{1}{2}} < \sqrt{8\pi}$ is satisfied. Then the following statements hold: 1) There exi a function $Q \in \bigcap_{q < \infty} L^q(\Sigma)$, and a number $\lambda \in \mathbb{R}$ such tha

Theorems & Definitions (45)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • Theorem 4: Theorems 1.1 and 1.2 of kuwert2015two
  • Remark 5
  • Lemma 6
  • Proposition 7
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 35 more