1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

2019/05/14 by Hu, Yingxiang, Xu, Shicheng
#53C20 #53C21 #53C24 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1905.05572

Abstract

Let Mⁿ be a closed convex hypersurface lying in a convex ball B(p,R) of the ambient (n+1)-manifold Nⁿ⁺¹. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of B(p,R), 1st eigenvalue and mean curvature of M, not only M is Hausdorff close and almost isometric to a geodesic sphere S(p₀,R₀) in N, but also its enclosed domain is C^1,α-close to a geodesic ball of constant curvature.

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