Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

  • Ben Andrews

    Australian National University, Canberra, Australia
  • Xuzhong Chen

    Hunan University, Changsha, China
  • Yong Wei

    University of Science and Technology of China, Hefei, China
Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space cover
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Abstract

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of kk-th mean curvatures with k=1,,nk=1,\ldots,n, and positive powers of pp-th power sums SpS_p with p>0p > 0. We prove that if the initial hypersurface M0M_0 is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time t>0t > 0, exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature.

In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function ff of the shifted principal curvatures λi=κi1\lambda_i=\kappa_i-1, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where ff is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

Cite this article

Ben Andrews, Xuzhong Chen, Yong Wei, Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space. J. Eur. Math. Soc. 23 (2021), no. 7, pp. 2467–2509

DOI 10.4171/JEMS/1059