Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

TL;DR

This paper studies convex hypersurfaces with pinched principal curvatures, proving bounds on geometric ratios and analyzing their evolution under curvature-driven flows, showing they become spherical as they contract.

Contribution

It establishes new bounds on the ratio of circumradius to inradius for hypersurfaces with pinched curvatures and analyzes their flow under high-power curvature functions.

Findings

Bound on the ratio of circumradius to inradius depending on the circumradius.

Convex hypersurfaces evolve to spherical shapes under curvature flows.

Solutions remain smooth and strictly convex until contraction to a point.

Abstract

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time.