Locally constrained curvature flows and geometric inequalities in hyperbolic space

Abstract

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li (An inverse curvature type hypersurface flow in ${\mathbb {H}}^{n+1}$, preprint). This flow preserves the mth quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov–Fenchel type inequalities in hyperbolic space. Some special cases have been studied in Brendle et al. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of kth Gauss–Bonnet curvature of a smooth h-convex hypersurface to its mth quermassintegral (for $0\le m\le 2k+1\le n$), and comparing the weighted integral of kth mean curvature to its mth quermassintegral (for $0\le m\le k\le n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu (Math Z 281, 257–297, 2015). In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of kth shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang’s (Commun Pure Appl Math 69(1), 124–144, 2016) inequality.