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Stability of the Euclidean 3-ball under L2-curvature pinching

Olivier Graf

Abstract

In this article, we consider compact Riemannian 3-manifolds with boundary. We prove that if the $L^2$-norm of the curvature is small and if the $H^{1/2}$-norm of the difference of the fundamental forms of the boundary is small, then the manifold is diffeomorphic to the Euclidean ball. Moreover, we obtain that the manifold and the ball are metrically close (uniformly and in $H^2$-norm), with a quantitative, optimal bound. The required smallness assumption only depends on the volumes of the manifold and its boundary and on a trace and Sobolev constant of the manifold. The proof only relies on elementary computations based on the Bochner formula for harmonic functions and tensors, and on the 2-spheres effective uniformisation result of Klainerman-Szeftel.

Stability of the Euclidean 3-ball under L2-curvature pinching

Abstract

In this article, we consider compact Riemannian 3-manifolds with boundary. We prove that if the -norm of the curvature is small and if the -norm of the difference of the fundamental forms of the boundary is small, then the manifold is diffeomorphic to the Euclidean ball. Moreover, we obtain that the manifold and the ball are metrically close (uniformly and in -norm), with a quantitative, optimal bound. The required smallness assumption only depends on the volumes of the manifold and its boundary and on a trace and Sobolev constant of the manifold. The proof only relies on elementary computations based on the Bochner formula for harmonic functions and tensors, and on the 2-spheres effective uniformisation result of Klainerman-Szeftel.

Paper Structure

This paper contains 20 sections, 32 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Remarks on Theorem \ref{['thm:main']}.
  4. Elliptic PDE heuristic.
  5. From elliptic estimates to geometric conclusions: an example.
  6. Outline of the proof of Theorem \ref{['thm:main']}.
  7. Related works.
  8. Applications to General Relativity.
  9. Acknowledgements.
  10. The harmonic radius vectorfield $X$
  11. Functional estimates
  12. Sobolev estimates
  13. Elliptic estimates on ${{\partial}M}$
  14. Elliptic estimates on $M$
  15. Uniformisation of ${\partial}M$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Lambda>0$. There exists $\varepsilon^0>0$, depending only on $\Lambda$, such that for all $0<\varepsilon<\varepsilon^0$ the following holds. Any smooth oriented connected and compact 3-dimensional Riemannian manifold $(M,g)$ with non-empty boundary, such that and satisfies the following properties:

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Theorem
  • proof
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 62 more