A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^\alpha f^{-\beta}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $\alpha \le 0<\beta\le 1-\alpha$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt \cite{GC3} and Urbas \cite{UJ2} can be recovered by putting $\alpha=0$ and $\beta=1$ in our first result. If the initial hypersurface is convex, this is our previous work \cite{DL}. If $\alpha \le 0<\beta< 1-\alpha$ and the ambient space is hyperbolic space $\mathbb{H}^{n+1}$, we prove that the flow $\frac{\partial X}{\partial t}=(u^\alpha f^{-\beta}-\eta u)\nu$ has a longtime existence and smooth convergence to a coordinate slice. The flow in $\mathbb{H}^{n+1}$ is equivalent (up to an isomorphism) to a re-parametrization of the original flow in $\mathbb{R}^{n+1}$ case. Finally, we find a family of monotone quantities along the flows in $\mathbb{R}^{n+1}$. As applications, we give a new proof of a family of inequalities involving the weighted integral of $k$th elementary symmetric function for $k$-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in \cite{GL2}.