Gauss curvature flow with shrinking obstacle

TL;DR

This paper studies the evolution of shapes driven by Gauss curvature with an obstacle constraint, establishing optimal curvature bounds, long-term existence, and regularity of free boundaries across all dimensions.

Contribution

It introduces a comprehensive analysis of Gauss curvature flow with obstacles, proving optimal bounds, existence, and boundary regularity for all dimensions and positive powers.

Findings

Established optimal curvature bounds for the flow.

Proved long-time existence of solutions.

Demonstrated $C^1$ regularity of free boundaries.

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.