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

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 $k=1,\cdots ,n$, 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 $M_t$ 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 ${\lambda }_i={\kappa }_i-1$, 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.

42 pages, 2 figures

References in corpus (2)

• Fully nonlinear parabolic equations in two space variables
• Hypersurfaces in H^{n+1} and conformally invariant equations: the generalized Christoffel and Nirenberg problems

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• Contraction of surfaces in hyperbolic space and in sphere
• Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space
• The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space
• Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow
• The quermassintegral preserving mean curvature flow in the sphere
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