Inverse Gauss curvature flows with free boundaries in a cone

TL;DR

This paper studies the evolution of convex hypersurfaces with free boundaries in a cone under Gauss curvature flows, proving long-term existence and convergence to a spherical shape after rescaling.

Contribution

It establishes the long-time existence and smooth convergence of convex hypersurfaces with free boundaries in a cone driven by Gauss curvature power, a new result in geometric flow theory.

Findings

Hypersurfaces expand inside the cone under the flow.

The flow exists for all time and converges smoothly.

Rescaled hypersurfaces approach a round sphere.

Abstract

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere after rescaling.