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 $\alpha \ge \beta +1$ 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 $r^{\alpha }{F}^{\beta }$.