Pinched hypersurfaces contract to round points

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 $n\in \mathbb{N}$.

14 pages

References in corpus (2)

• Surfaces moving by powers of Gauss curvature
• When maximum-principle functions cease to exist