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Optimal quantitative stability of the Möbius group of the sphere in all dimensions

André Guerra, Xavier Lamy, Konstantinos Zemas

Abstract

In any dimension $n\geq 3$, we prove an optimal stability estimate for the Möbius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{λ>0,φ\in \mathrm{Möb}(\mathbb S^{n-1})} \int_{\mathbb S^{n-1}}\left|\frac 1λ\nabla_{T} u -\nabla_{ T}φ\right|^{n-1} d\mathcal H^{n-1} \leq C_n \mathcal E_{n-1}(u).$ Here, $\mathcal E_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u(\mathbb S^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\geq 4$.

Optimal quantitative stability of the Möbius group of the sphere in all dimensions

Abstract

In any dimension , we prove an optimal stability estimate for the Möbius group among maps , of the form Here, is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions .
Paper Structure (22 sections, 27 theorems, 269 equations)

This paper contains 22 sections, 27 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm:compactintro']} and relation to H-systems.
  4. Bubbling analysis for $n=3$
  5. The higher dimensional case $n>3$: difficulties
  6. The higher dimensional case $n>3$: proof sketch
  7. Quantitative stability and the proof of Theorem \ref{['thm:qttiveintro']}
  8. Outline of the paper
  9. Notation
  10. Preliminaries
  11. Multilinear algebra and calculus on the sphere
  12. On the topological degree and the volume functional
  13. Möbius transformations
  14. The first and second variations of the deficit
  15. The parametric conformal isoperimetric inequality
  16. ...and 7 more sections

Key Result

Proposition 1.1

For all $u\in W^{1,n-1}(\mathbb S^{n-1};\mathbb{R}^n)$ we have with equality if and only if $(u-y_0)/|\mathcal{V}_n(u)|^{1/n}\in \textup{M\"ob}(\mathbb S^{n-1})$ for some $y_0\in \mathbb{R}^n$.

Theorems & Definitions (50)

  • Proposition 1.1: Conformal isoperimetric inequality
  • Theorem 1.2: Compactness
  • Theorem 1.3: Optimal quantitative stability
  • Corollary 1.4: Stability for sphere-valued maps
  • Remark 1.5: Optimality
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4: Comparison with VMO-degree
  • ...and 40 more