Surfaces expanding by non-concave curvature functions

TL;DR

This paper studies the evolution of convex surfaces in different space forms expanding by a curvature-dependent flow, proving long-term existence and convergence without requiring the curvature function to be concave, and analyzing the shape of the limit surface.

Contribution

It establishes long-time existence and convergence of convex surface flows in space forms with non-concave curvature functions, extending previous results.

Findings

Flow preserves the pinching ratio of the initial surface.

Long-time existence and convergence of the flow are proven.

Limit shapes may not be round in hyperbolic space for certain parameters.

Abstract

In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(\kappa)~(\kappa=0,1,-1)$ expanding by $F^{-\alpha}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $\alpha\in(0,1]$ for $\kappa=0,-1$ and $\alpha=1$ for $\kappa=1$. By deriving that the pinching ratio of the flow surface $M_t$ is no greater than that of the initial surface $M_0$, we prove the long time existence and the convergence of the flow. No concavity assumption of $F$ is required. We also show that for the flow in $\mathbb{H}^3$ with $\alpha\in (0,1)$, the limit shape may not be necessarily round after rescaling.