Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space

Abstract

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space H n , by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.2010 Mathematics Subject Classification. 52A40, 53C65, 53C44. Key words and phrases. quermassintegral, curvature integral, isoperimetric problem, Alexandrov-Fenchel inequality.GW is partly supported by SFB/TR71 "Geometric partial differential equations" of DFG. Part of this work was done while CX was visiting the mathematical institute of Albert-Ludwigs-Universität Freiburg. He would like to thank the institute for its hospitality.

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

“…We note that, after this paper was written, several related inequalities for hypersurfaces in hyperbolic space have appeared in the literature; see, e.g., [19,24]. 1 2 Star-Shaped Hypersurfaces in the AdS-Schwarzschild Manifold LEMMA 2.1.…”

Quermassintegral preserving curvature flow in Hyperbolic space

“…To describe the global term φ(t) in (1.1), we first recall the (normalized) k-th mean curvature E k of a smooth closed hypersurface M and the quermassintegrals W k (Ω) of the bounded domain Ω enclosed by M : If Ω is a (geodesically) convex domain in H n+1 , then the quermassintegrals of Ω are defined as follows (see [30,31,33]):…”

Locally constrained inverse curvature flows

“…The most prominent example of an expanding flow is the inverse mean curvature flow, a weak notion of which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality, [32]. Various other applications of contracting and expanding flows include a classification of 2-convex n-dimensional hypersurfaces using the mean curvature flow with surgery, due to Huisken and Sinestrari for n ≥ 3, [34], various extensions of geometric inequalities of Alexandrov-Fenchel-type to nonconvex hypersurfaces, [8], [25], new Alexandrov-Fenchel-type inequalities in the hyperbolic space [15,44,45] and in the sphere [12,24,36,45].…”

“…A full convergence result for closed, starshaped and k-convex initial hypersurfaces would prove the quermass Alexandrov-Fenchel inequalities for such hypersurfaces. For horo-convex domains these have been established by Wang and the second author [44] using a global quermassintegral preserving curvature flow. 1 A hypersurface in the hyperbolic space is called horo-convex if all its principal curvatures are greater or equal than 1.…”