Xianfeng Wang

In this paper, we investigate closed strictly convex hypersurfaces in ${\mathbb{R}}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree $1$ and inverse concave function of the principal curvatures with power greater than $1$, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere ${\mathbb{S}}_{+}^{n+1}$ with power greater than or equal to $1$.

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\le 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in ${\mathbb{R}}^{n+1}$ and ${\mathbb{S}}^{n+1}$ by ${\sigma }_k^{\alpha }$, where ${\sigma }_k$ is the $k$-th elementary symmetric function of the principal curvatures and $\alpha \ge 1/k$. We prove that for any $n\ge 3$ and any fixed $k$ with $1\le k\le n$, there exists a constant $c(n,k)>1/k$ such that that if $\alpha$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $1<k\le n\le k^2$, we can choose $c(n,k)=\frac{1}{k-1}$. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.