Small L2 curvature plus boundary second fundamental form close to the unit sphere forces a 3-manifold with boundary to be diffeomorphic to, and quantitatively close to, the Euclidean ball.
Global nonlinear stability of Minkowski space for spacelike-characteristic initial data
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abstract
In this paper, we prove the global nonlinear stability of Minkowski space in the context of the spacelike-characteristic Cauchy problem for Einstein vacuum equations. Spacelike-characteristic initial data are posed on a compact 3-disk and on the future complete null hypersurface emanating from its boundary. Our result extends the seminal stability result for Minkowski space proved by Christodoulou and Klainerman for which initial data are prescribed on a spacelike 3-plane. The proof relies on the classical vectorfield method and bootstrapping argument from Christodoulou-Klainerman. The main novelty is the introduction and control of new geometric constructions adapted to the spacelike-characteristic setting. In particular, it features null cones with prescribed vertices, spacelike maximal hypersurfaces with prescribed boundaries and global harmonic coordinates on Riemannian 3-disks.
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math.DG 1years
2025 1verdicts
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Stability of the Euclidean 3-ball under L2-curvature pinching
Small L2 curvature plus boundary second fundamental form close to the unit sphere forces a 3-manifold with boundary to be diffeomorphic to, and quantitatively close to, the Euclidean ball.