A class of curvature flows expanded by support function and curvature function

TL;DR

This paper studies a class of curvature flows for convex hypersurfaces in Euclidean space, proving long-term existence and convergence to a sphere under specific conditions on the flow parameters.

Contribution

It introduces a new class of expanding curvature flows involving support and curvature functions, establishing existence, uniqueness, and convergence results.

Findings

Flow exists for all time and remains smooth and convex.

Normalized flow converges smoothly to a sphere.

Results hold under specific parameter conditions lpha <eta -lpha.

Abstract

In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbb{R}^{n+1} with speed u^\alpha f^\beta (\alpha, \beta\in\mathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If \alpha \leq 0<\beta\leq 1-\alpha, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.