Pinched hypersurfaces contract to round points

TL;DR

This paper studies the evolution of convex hypersurfaces under curvature-driven flows, proving convergence to round points for 2-pinched initial shapes and extending results to higher dimensions with new quantities.

Contribution

It introduces new quantities for proving convergence in arbitrary dimensions and establishes convergence results for various curvature flows.

Findings

2-pinched hypersurfaces contract to round points

Convergence results extended to higher dimensions

New quantities introduced for analysis in arbitrary dimensions

Abstract

We investigate the evolution of closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$, n=3, for contracting normal velocities, including powers of the mean curvature, of the norm of the second fundamental form, and of the Gauss curvature. We prove convergence to a round point for 2-pinched initial hypersurfaces. In $\mathbb{R}^{n+1}$, n=2, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\mathbb{N}$.