Explicit rigidity of almost-umbilical hypersurfaces

TL;DR

This paper provides explicit estimates for how close almost-umbilical hypersurfaces are to spheres, using integral norms of their traceless second fundamental form, and determines decay rates via inverse mean curvature flow.

Contribution

It introduces explicit quantitative bounds relating hypersurface closeness to spheres with small traceless second fundamental form and analyzes decay rates in Euclidean space using inverse mean curvature flow.

Findings

Closeness to a sphere is controlled by a power of the integral norm of the traceless second fundamental form.

Established the best possible decay order for hypersurfaces in Euclidean space.

Provided explicit estimates for hypersurface proximity to geodesic spheres.

Abstract

We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Furthermore we use the inverse mean curvature flow in the hyperbolic space to deduce the best possible order of decay in the class of $C^{\infty}$-bounded hypersurfaces of the Euclidean space.