1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

Yingxiang Hu; Shicheng Xu

paper

1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

arXiv:1905.05572

Abstract

Let $M^{n}$ be a closed convex hypersurface lying in a convex ball $B (p, R)$ of the ambient $(n + 1)$ -manifold $N^{n + 1}$ . 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 $S (p_{0}, R_{0})$ in $N$ , but also its enclosed domain is $C^{1, α}$ -close to a geodesic ball of constant curvature.

6 pages