1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

Hu, Yingxiang; Xu, Shicheng

doi:10.1090/proc/14916

1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold
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by Yingxiang Hu and Shicheng Xu PDF

Proc. Amer. Math. Soc. 148 (2020), 2609-2615 Request permission

Abstract:

This is an application of our previous quantitative rigidity result via pinching Heintze–Reilly’s inequality. Based on work by Hu and Xu [Recognizing shape via 1st eigenvalue, mean curvature, and upper curvature bound, arXiv:1905.01664v2], we prove that for any closed convex hypersurface $M^n$ lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$, whose sectional curvature $\mu \le K_N\le \delta$, if $\lambda _1(M)$ approaches $n(\delta +\|H\|_\infty ^2)$, then $M$ (resp., its enclosed domain) is Hausdorff (resp., $C^{1,\alpha }$) close to a sphere (resp., a geodesic ball) of constant curvature, where $\lambda _1(M)$ is the 1st eigenvalue of $M$ and $\|H\|_\infty$ is the maximum of $M$’s mean curvature in $N$.

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