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

TL;DR

This paper proves that compact 3-manifolds with boundary are close to Euclidean balls if their curvature and boundary forms are sufficiently small in specific norms, using elementary methods and uniformization results.

Contribution

It establishes a stability result for the Euclidean 3-ball under L2-curvature pinching with explicit bounds and elementary proofs.

Findings

Manifolds with small L2-curvature are diffeomorphic to the Euclidean ball.

Quantitative bounds relate manifold closeness to curvature and boundary form norms.

The approach relies on elementary computations and uniformization techniques.

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.