Abstract
In this paper, we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold. We give C0, C1, C2 estimates of the flow. Using these facts, we prove that the solution exists for all time and the principal curvatures converge to 1 polynomially fast.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brendle S, Hung P-K, Wang M-T. A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold. Comm Pure Appl Math, 2016, 69: 124–144
Chen L, Mao J, Tu Q, et al. Asymptotic convergence for a class of inverse mean curvature flows in ℝn+1. Proc Amer Math Soc, 2020, 148: 379–392
Gerhardt C. Flow of nonconvex hypersurfaces into spheres. J Differential Geom, 1990, 32: 299–314
Gerhardt C. Curvature Problems. Series in Geometry and Topology, vol. 39. Somerville: International Press, 2006
Gerhardt C. Inverse curvature flows in hyperbolic space. J Differential Geom, 2011, 89: 487–527
Guan P, Li J. The quermassintegral inequalities for k-convex starshaped domains. Adv Math, 2009, 221: 1725–1732
Huisken G. Flow by mean curvature of convex surfaces into spheres. J Differential Geom, 1984, 20: 237–266
Huisken G. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent Math, 1986, 84: 463–480
Krylov N V. Nonlinear Elliptic and Parabolic Equations of the Second Order. Netherlands: Dordrecht, 1987
Li H, Wei Y, Xiong C. A geometric inequality on hypersurface in hyperbolic space. Adv Math, 2014, 253: 152–162
Li Q-R, Sheng W M, Wang X-J. Asymptotic convergence for a class of fully nonlinear curvature flows. J Geom Anal, 2020, 30: 834–860
Scheuer J. Inverse curvature flows in Riemannian warped products. J Funct Anal, 2019, 276: 1097–1144
Urbas J. On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math Z, 1990, 205: 355–372
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11831005) and a collaboration project funded by National Natural Science Foundation of China and the Research Foundation Flanders (Grant No. 11961131001).
Author information
Authors and Affiliations
Corresponding author
Additional information
In Memory of Professor Zhengguo Bai (1916–2015)
Rights and permissions
About this article
Cite this article
Li, H., Xu, B. A class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold. Sci. China Math. 64, 1573–1588 (2021). https://doi.org/10.1007/s11425-020-1833-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-020-1833-2