Closeness to spheres of hypersurfaces with normal curvature bounded below

TL;DR

This paper derives geometric estimates for hypersurfaces with a lower bound on normal curvature in Riemannian manifolds, relating their shape to distances and angles relative to a fixed interior point.

Contribution

It provides new bounds on the closeness of such hypersurfaces to spheres, based on curvature and distance measurements in Riemannian manifolds.

Findings

Bounds on the angle between geodesics and normals

Estimates for the width of spherical shells containing the hypersurface

Quantitative relations between curvature bounds and geometric proximity

Abstract

For a Riemannian manifold $M^{n+1}$ and a compact domain $\Omega \subset M^{n+1}$ bounded by a hypersurface $\partial \Omega$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial \Omega$ for the angle between the geodesic line joining a fixed interior point $O$ in $\Omega$ to a point on $\partial \Omega$ and the outward normal to the surface. Estimates for the width of a spherical shell containing such a hypersurface are also presented.