Quermassintegral preserving curvature flow in Hyperbolic space

Abstract

We consider the quermassintegral preserving flow of closed h-convex hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function f of the principal curvatures which is inverse concave and has dual f * approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is h-convex, then the solution of the flow becomes strictly h-convex for t > 0, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

“…Without resorting to the constant rank theorem, here we follow the spirit of Andrews-Wei [8], and prove that the h-convexity is preserved along the IMCF. Theorem 9.…”

“…In Section 4, we use Andrews' maximum principle for tensors to show that if the initial hypersurface Σ in hyperbolic space is h-convex, then the flow hypersurface Σ t of the IMCF becomes strictly h-convex for t > 0. The idea we used here follows from the recent work of Andrews and Wei [8], and the proof does not rely on the constant rank theorem as in [31]. This property will be crucial to establish the rigidity part of the inequality (1.5) under the weaker condition that Σ has nonnegative sectional curvature.…”

New Pinching Estimates for Inverse Curvature Flows in Space Forms

“…As the Euclidean case in [3,7,29], we need to derive a parabolic equation of the support function via the Gauss map parametrization which is concave with respect to the second spatial derivatives. Such Gauss map parametrization of curvature flows in hyperbolic space has been formulated recently by the author with Andrews [8] to study the quermassintegral preserving curvature flow in hyperbolic space.…”

“…Firstly, we briefly review the Gauss map parametrization of curvature flows in hyperbolic space briefly and refer the readers to [8] for details. Denote by R 1,n+1 the Minkowski spacetime, that is the vector space R n+2 endowed with the Minkowski spacetime metric ·, · by…”

“…In the hyperbolic space and in sphere, we can also develop the similar idea. In [8], the author with Andrews derived the C 2,α regularity of the volume preserving flow in the hyperbolic space by writing the flow as a scalar parabolic PDE of the support function of the corresponding evolution in the unit Euclidean ball, which is concave with respect to the second spatial derivatives.…”