Regularity of the $p$-Gauss curvature flow with flat side

TL;DR

This paper investigates the smoothness properties of convex hypersurfaces evolving under the $p$-Gauss curvature flow, focusing on the behavior near flat sides, building on previous work on interface regularity.

Contribution

It extends prior results by analyzing the regularity of the entire convex hypersurface near the flat interface in the $p$-Gauss curvature flow.

Findings

Established regularity of the hypersurface near the flat interface.

Extended understanding of the geometric evolution under the $p$-Gauss curvature flow.

Provided new techniques for analyzing boundary regularity in curvature flows.

Abstract

We study the regularity of the $p$-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of the convex hypersurface near the interface.