Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

TL;DR

This paper proves that certain fully nonlinear contracting curvature flows of convex hypersurfaces in Euclidean space exist globally and converge exponentially to spheres, with detailed analysis of conditions on the speed function and special curvature cases.

Contribution

It establishes convergence results for a broad class of nonlinear curvature flows, including new asymptotic behavior and counterexamples under specific parameter conditions.

Findings

Flow exists for all time under specified conditions.

Normalized flow converges exponentially to a sphere.

Counterexample shows non-convergence when parameters do not meet conditions.

Abstract

In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with the normal speed $\Phi$ given by $r^\alpha F^\beta$ or $u^\alpha F^\beta$, where $F$ is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, $r$ is the distance from the hypersurface to the origin and $u$ is the support function of hypersurface. If $\alpha\geq \beta+1$ when $\Phi=r^\alpha F^\beta$ or $\alpha> \beta+1$ when $\Phi=u^\alpha F^\beta$, we prove that the flow exists for all times and converges to the origin. After proper rescaling, we prove that the normalized flow converges exponentially in the $C^\infty$ topology to a sphere centered at the origin. Furthermore, for special inverse concave curvature function $F=K^{\frac{s}{n}}F_1^{1-s}(s\in(0, 1])$, where $K$ is Gauss curvature and $F_1$ is inverse-concave, we obtain the asymptotic convergence for the flow with $\Phi=u^\alpha F^\beta$ when $\alpha=\beta+1$. If $\alpha<\beta+1$, a counterexample is given for the above convergence when speed equals to $r^\alpha F^\beta$.