Gauss curvature flow with shrinking obstacle

Abstract

We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the $C^1$ regularity of free boundaries under a uniform thickness assumption.

To appear in Math. Ann