Surfaces moving by powers of Gauss curvature

Abstract

We prove that strictly convex surfaces moving by $K^{\alpha /2}$ become spherical as they contract to points, provided $\alpha$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for surfaces moving by arbitrary functions of curvature, by finding a quantity conserved by the reaction terms in the evolution of curvature.

References in corpus (2)

• Positively curved surfaces in the three-sphere
• Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

Cited by in corpus (4)

• On maximum-principle functions for flows by powers of the Gauss curvature
• When maximum-principle functions cease to exist
• Pinched hypersurfaces contract to round points
• An application of dual convex bodies to the inverse Gauss curvature flow