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

TL;DR

This paper studies a class of curvature flows for star-shaped hypersurfaces in Euclidean and hyperbolic spaces, proving long-term existence, convergence to spheres or slices, and deriving new geometric inequalities.

Contribution

It introduces a generalized curvature flow involving support and curvature functions, extending previous results and establishing convergence and inequality results in both Euclidean and hyperbolic spaces.

Findings

Flow exists for all time and converges smoothly to a sphere in Euclidean space.

Flow exists for all time and converges to a coordinate slice in hyperbolic space.

New geometric inequalities involving weighted integrals of elementary symmetric functions.

Abstract

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}.