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

2018/05/30 by Ben Andrews, Andrews, Ben, Xuzhong Chen +3 · 2 citations
Mathematics · #53C21 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1805.11776

openalex publication_date 2018/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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,⋯,n, and positive powers of p-th power sums S_p with p>0. We prove that if the initial hypersurface M₀ is smooth and closed and has positive sectional curvatures, then the solution M_t 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 homogeneous degree one function f of the shifted principal curvatures λ_i=κ_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.

Cited by

• The quermassintegral inequalities for horo-convex domains in the sphere
• New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

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