Yusha Lv

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 $r^{\alpha }{F}^{\beta }$.

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $\Phi$ is given by a power $\beta \ge 1$ of a monotone symmetric and homogeneous of degree one function $F$ of the principal curvatures. Under the assumption that $F$ is inverse concave and its dual function approaches zero on the boundary of positive cone, we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence. In particular, if $F=K^{s/n}{G}^{1-s}$ for any $s\in (0,1]$, where $G$ is a homogeneous of degree one, increasing in each argument and inverse concave curvature function, we prove that the complete noncompact smooth strictly convex solution exists and remains a graph for all times.