Asymptotic convergence for a class of anisotropic curvature flows

TL;DR

This paper investigates the long-term behavior of a class of anisotropic curvature flows for star-shaped hypersurfaces, proving convergence to spheres under various conditions and generalizing previous results to broader classes.

Contribution

It introduces new auxiliary functions to analyze the flow and extends convergence results from convex to $k$-convex hypersurfaces, broadening the scope of prior work.

Findings

Flow solutions exist for all time under specified conditions.

Solutions converge smoothly to spheres after normalization.

Generalizes previous convergence results to broader classes of hypersurfaces.

Abstract

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\frac{\alpha}{\beta}}\sigma_k^{\frac{1}{\beta}}$, where $\sigma_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $\alpha$, $\beta$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $\alpha$, $\beta$. When $k\geq2$, $0<\beta\leq 1$, $\alpha\geq \beta+k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to $k$-convex. When $k \geq 2$, $\beta=k$, $\alpha \geq 2k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from $k=2$ to $k \geq 2$.