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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.03823 |
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Table of Contents:
- 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.