Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

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 $r^\alpha F^\beta$.