Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

TL;DR

This paper studies the long-term behavior of a class of inverse curvature flows of convex hypersurfaces in a cone, proving smooth convergence to a spherical shape under certain conditions.

Contribution

It establishes the long-time existence and smooth convergence of inverse curvature flows in a cone for a broad class of speed functions, extending previous results.

Findings

Flow exists for all time under given conditions.

Hypersurfaces converge smoothly to a sphere.

Results apply to a class of nonlinear inverse curvature flows.

Abstract

For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to $\left(f(r)H\right)^{-1}$, where $f$ is a positive function of the radial distance parameter $r$ and $H$ is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function $f$, we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere.