Motion of hypersurfaces by Gauss curvature

Abstract

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-smooth initial hypersurfaces, and develop upper and lower bounds on the speed and the curvature independent of initial conditions. Applications are given to the flow by affine normal and to the existence of non-spherical homothetically contracting solutions. Show more