Recognizing shape via 1st eigenvalue, mean curvature and upper curvature bound

Yingxiang Hu, Shicheng Xu

Published 2019-05-05Version 1

Let $M^n$ be a closed immersed hypersurface lying in a contractible 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 to a geodesic sphere $S(p_0,R_0)$ in $N$, but also the ``enclosed'' ball $B(p_0,R_0)$ is close to be of constant curvature, provided with a uniform control on the volume and mean curvature of $M$. We raise a conjecture for $M$ to be a diffeomorphic sphere, and give positive partial answers for several special cases, one of which is as follows: If in addition, the renormalized $L^q$ norm ($q>n$) of $M$'s 2nd fundamental form admits a uniform upper bound, then $M$ is an embedded diffeomorphic sphere, almost isometric to $S(p_0, R_0)$, and intrinsically $C^{\alpha}$-close to a round sphere of constant curvature.