Hypersurfaces with small extrinsic radius or large λ₁ in Euclidean spaces

2010/09/10 by Erwann Aubry, Aubry, Erwann, Jean-François Grosjean +4 · 1 citation
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.1009.2010

24 pages

openalex publication_date 2010/09/10 · arxiv created 2010/11/25 · arxiv updated 2010/11/29 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28

Abstract

We prove that hypersurfaces of \Rⁿ⁺¹ which are almost extremal for the Reilly inequality on λ₁ and have L^p-bounded mean curvature (p>n) are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that when a supplementary L^q bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when q>n, but not necessarily diffeomorphic to a sphere when q\leqslant n.

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• Metric shape of hypersurfaces with small extrinsic radius or large λ₁

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