Abstract
In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface \(\Sigma \) is strictly mean convex and star-shaped, then the flow hypersurface \(\Sigma _t\) converges to a large coordinate sphere as \(t\rightarrow \infty \) exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler–Schwarzschild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken–Ilmanen’s (J Differ Geom 80:433–451, 2008) result.
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References
Bray, H., Neves, A.: Classification of prime 3-manifolds with Yamabe invariant greater than \({\mathbb{RP}}^3\). Ann. Math. 159, 407–424 (2004)
Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publications Mathématiques de l’IHÉS 117, 247–269 (2013)
Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski-type inequality for hypersurfaces in the Anti-deSitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)
De lima, L.L., Girão, F.: An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a penrose inequality. Ann. Henri Poincaré 17(4), 979–1002 (2016)
De Lima, L.L., Girão, F.: A Penrose inequality for asymptotically locally hyperbolic graphs. arXiv:1304.7887
Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32, 299–314 (1990)
Gerhardt, C.: Curvature Problems, Ser. in Geom. and Topol., vol. 39. International Press, Somerville (2006)
Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487–527 (2011)
Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geom. 100, 301–347 (2014)
Guan, P., Li, J.: The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221, 1725–1732 (2009)
Ge, Y., Wang, G., Wu, J.: Hyperbolic Alexandrov–Fenchel quermassintegral inequalities II. J. Differ. Geom. 98(2), 237–260 (2014)
Ge, Y., Wang, G., Wu, J., Xia, C.: A Penrose inequality for graphs over Kottler space. Calc. Var. PDE 52, 755–782 (2015)
Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)
Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. math. 84(3), 463–480 (1986)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–438 (2001)
Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80, 433–451 (2008)
Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel, Dordrecht (1987)
Lee, D.A., Neves, A.: The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass. Commun. Math. Phys. 339, 327–352 (2015)
Li, H., Wei, Y., Xiong, C.: A geometric inequality on hypersurface in hyperbolic space. Adv. Math. 253, 152–162 (2014)
Li, H., Wei, Y., Xiong, C.: Erratum: A note on Weingarten hypersurface in the warped product manifold. Intern. J. Math 27(13), 1692001 (2016)
Makowski, M., Scheuer, J.: Rigidity results, inverse curvature flows and Alexandrov–Fenchel type inequalities in the sphere. Asian J. Math. 20(5), 869–892 (2016)
Neves, A.: Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds. J. Differ. Geom. 84(1), 191–229 (2010)
Petersen, P.: Riemannian Geometry, GTM 171, 2nd edn. Springer, New York (2006)
Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)
Urbas, J.: On the expansion of star-shaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205, 355–372 (1990)
Wei, Y., Xiong, C.: Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere. Pacific J. Math. 277(1), 219–239 (2015)
Acknowledgements
The authors would like to thank the referee for carefully reading and useful comments. The first author was supported by NSFC No. 11671224, the second author was supported by Jason D. Lotay through his EPSRC Grant EP/K010980/1.
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Communicated by A. Neves.
