Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

TL;DR

This paper investigates the long-term behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows, proving convergence to a sphere for star-shaped, k-convex hypersurfaces under certain conditions.

Contribution

It generalizes previous results by establishing long-time existence and smooth convergence of the flow to a sphere for a broader class of curvature flows involving a non-homogeneous factor.

Findings

Flow exists for all time for star-shaped, k-convex hypersurfaces.

Flow converges smoothly to a sphere after normalization.

Generalizes previous asymptotic results for specific curvature flows.

Abstract

In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its speed $f(r)\sigma_k^\alpha$ where $\sigma_k$ is the $k$-th elementary symmetric polynomial of $M$'s principle curvatures, $r$ is the distance of the point on $M$ to the origin, $f$ is a smooth nonnegative function on $[0,\infty)$ and $\alpha > 0$. Under some suitable conditions on $f$, we prove that starting from a star-shaped and $k$-convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in \cite{li2022asymptotic}.