Locally constrained curvature flows and geometric inequalities in hyperbolic space
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Mathematische Annalen
Locally constrained curvature flows and geometric inequalities in hyperbolic space Yingxiang Hu1 · Haizhong Li2 · Yong Wei3 Received: 28 February 2020 / Revised: 26 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 Hn+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 ≤ m ≤ 2k + 1 ≤ n), and comparing the weighted integral of kth mean curvature to its mth quermassintegral (for 0 ≤ m ≤ k ≤ 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. Mathematics Subject Classification 53C44 · 52A39
Communicated by Y. Giga. Extended author information available on the last page of the article
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . 1.1 Brendle–Guan–Li’s flow and applications 1.2 New locally constrained curvature flow . . 2 Preliminaries . . . . . . . . . . . . . . . . . . 2.1 Elementary symmetric functions . . . . . 2.2 Hypersurfaces in hyperbolic space . . . . 2.3 Parametrization by radial graph . . . . . . 3 Evolution equations . . . . . . . . . . . . . . 4 Preserving h-convexity . . . . . . . . . . . . . 5 New geometric inequalities: I . . . . . . . . . 5.1 Proof of Theorem 1.3 . . . . . . . . . . . 5.2 Proof of Theorem 1.4 . . . . . . . . . . . 6 New locally constrained curvature flow . . . . 7 Curvature estimate . . . . . . . . . . . . . . . 7.1 Estimate on F . . . . . . . . . . . . . . . 7.2 Preserving h-convexity . . . . . . . . . . 7.3 Curvature estimate . . . . . . . . . . . . 8 Long time existence and convergence . . . .
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