\alpha-Gauss Curvature flows with flat sides

TL;DR

This paper investigates the evolution of convex surfaces in three-dimensional space under lpha-Gauss curvature flows, revealing smooth solutions, viscosity solutions, and a waiting time effect for flat sides, with interface regularity under certain conditions.

Contribution

It provides new results on existence, regularity, and interface behavior of solutions to lpha-Gauss curvature flows with flat sides for lpha in (1/2, 1].

Findings

Existence of smooth solutions for smooth, strictly convex initial data.

Viscosity solutions with $C^{1,1}$-estimates for convex initial data.

Persistence of flat sides for a positive waiting time.

Abstract

In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $\alpha$-power of positive part of Gauss Curvature. First, for 1/2<\alpha\leq 1$, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with $C^{1,1}$-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on $0 < t < T_0$ under certain necessary regularity and non-degeneracy initial conditions, where $ T_0$ is the vanishing time of the flat side.