Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

Xuzhong Chen,Ben Andrews,Yong Wei

doi:10.4171/jems/1059

Abstract

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,\ldots,n$, and positive powers of $p$-th power sums $S\_p$ with $p > 0$. We prove that if the initial hypersurface $M\_0$ is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time $t > 0$, exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function $f$ of the shifted principal curvatures $\lambda\_i=\kappa\_i-1$, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where $f$ is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

Highlights

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities
In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and homogeneous degree one function f of the shifted principal curvatures λi = κi − 1, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where f is assumed to satisfy a certain condition on the second derivatives
We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology

Summary

Introduction

In the second part of this paper, we will consider the flow of h-convex hypersurfaces in hyperbolic space with speed given by functions of the shifted Weingarten matrix W − I plus a global term chosen to preserve modified quermassintegrals of the evolving domains. To prove Theorem 1.7, we show that the inner radius and outer radius of the enclosed domain Ωt of the evolving hypersurface Mt satisfies a uniform estimate 0 < C−1 < ρ−(t) ≤ ρ+(t) ≤ C for some positive constant C This relies on the preservation of Wl(Ωt) and the monotonicity of Wl under the inclusion of h-convex domains stated in Proposition 1.6. Our inequalities (1.19) imply that the linear combinations of Wi(Ω) − fi ◦ f0−1(W0(Ω)) as in (1.22) are nonnegative for h-convex domains

Preliminaries

Preserving positive sectional curvature

Horospherically convex regions