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.
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Notes
This is possible since for any positive definite symmetric matrix A with \(A_{ij}\ge 0\) and \(A_{ij}v^iv^j=0\) for some \(v\ne 0\), there is a sequence of symmetric matrixes \(\{A^{(k)}\}\) approaching A, satisfying \(A^{(k)}_{ij}\ge 0\) and \(A^{(k)}_{ij}v^iv^j=0\) and with each \(A^{(k)}\) having distinct eigenvalues. Hence it suffices to prove the result in the case where all of \(\kappa _i\) are distinct.
\(F(\kappa )\) is inverse-concave, if its dual function \(F_*(z)=F(\frac{1}{z_1},\ldots ,\frac{1}{z_n})^{-1}\) is concave. As in [2], the quotient \(E_{m}(\kappa )/{E_{m-1}(\kappa )}\) is both concave and inverse concave.
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Acknowledgements
The authors would like to thank Professor Ben Andrews and Professor Pengfei Guan for helpful discussions, and the referees for their valuable comments and suggestions. Yingxiang Hu, was supported by China Postdoctoral Science Foundation (No. 2018M641317). Haizhong Li, was supported by NSFC Grant No. 11831005, 11671224 and NSFC-FWO Grant No. 1196131001. Yong Wei was supported by Discovery Early Career Researcher Award DE190100147 of the Australian Research Council and a research grant from University of Science and Technology of China.
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Hu, Y., Li, H. & Wei, Y. Locally constrained curvature flows and geometric inequalities in hyperbolic space. Math. Ann. 382, 1425–1474 (2022). https://doi.org/10.1007/s00208-020-02076-4
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DOI: https://doi.org/10.1007/s00208-020-02076-4