1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

TL;DR

This paper establishes that convex hypersurfaces in a convex ball of a Riemannian manifold, under eigenvalue and curvature pinching conditions, are geometrically close to geodesic spheres and balls of constant curvature.

Contribution

It proves a new eigenvalue pinching result that links spectral, geometric, and topological closeness of convex hypersurfaces to geodesic spheres in Riemannian manifolds.

Findings

Hypersurfaces are Hausdorff close to geodesic spheres.

Enclosed domains are $C^{1,eta}$-close to geodesic balls.

Eigenvalues and mean curvature are tightly controlled by curvature bounds.

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 $C^{1,\alpha}$-close to a geodesic ball of constant curvature.