Locally constrained inverse curvature flows

Scheuer, Julian; Xia, Chao

doi:10.1090/tran/7949

Locally constrained inverse curvature flows
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Trans. Amer. Math. Soc. 372 (2019), 6771-6803 Request permission

Abstract:

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order—namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti–de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.

References

Nicholas D. Alikakos and Alexandre Freire, The normalized mean curvature flow for a small bubble in a Riemannian manifold, J. Differential Geom. 64 (2003), no. 2, 247–303. MR 2029906
Ben Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. MR 1385524, DOI 10.1007/BF01191340
Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. MR 1267897
Ben Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. MR 2339467, DOI 10.1515/CRELLE.2007.051
Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 0485012
Simon Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247–269. MR 3090261, DOI 10.1007/s10240-012-0047-5
Simon Brendle, Pengfei Guan, and Junfang Li, An inverse curvature type hypersurface flow in $\mathbb {H}^{n+1}$ (private note).
Simon Brendle, Pei-Ken Hung, and Mu-Tao Wang, A Minkowski inequality for hypersurfaces in the anti–de Sitter–Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144. MR 3433631, DOI 10.1002/cpa.21556
Esther Cabezas-Rivas and Vicente Miquel, Volume preserving mean curvature flow in the hyperbolic space, Indiana Univ. Math. J. 56 (2007), no. 5, 2061–2086. MR 2359723, DOI 10.1512/iumj.2007.56.3060
Bennett Chow, Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. MR 826427
Bennett Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63–82. MR 862712, DOI 10.1007/BF01389153
Levi Lopes de Lima and Frederico Girão, An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality, Ann. Henri Poincaré 17 (2016), no. 4, 979–1002. MR 3472630, DOI 10.1007/s00023-015-0414-0
Klaus Ecker and Gerhard Huisken, Immersed hypersurfaces with constant Weingarten curvature, Math. Ann. 283 (1989), no. 2, 329–332. MR 980601, DOI 10.1007/BF01446438
Christian Enz, The scalar curvature flow in Lorentzian manifolds, Adv. Calc. Var. 1 (2008), no. 3, 323–343. MR 2458241, DOI 10.1515/ACV.2008.014
Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237–260. MR 3263518
Yuxin Ge, Guofang Wang, and Jie Wu, The GBC mass for asymptotically hyperbolic manifolds, Math. Z. 281 (2015), no. 1-2, 257–297. MR 3384870, DOI 10.1007/s00209-015-1483-y
Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314. MR 1064876
Claus Gerhardt, Closed Weingarten hypersurfaces in space forms, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 71–97. MR 1449403
Claus Gerhardt, Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. Reine Angew. Math. 554 (2003), 157–199. MR 1952172, DOI 10.1515/crll.2003.003
Claus Gerhardt, Curvature problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR 2284727
Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. MR 2879249
Claus Gerhardt, Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 471–489. MR 3148124, DOI 10.1007/s00526-012-0589-x
Claus Gerhardt, Curvature flows in the sphere, J. Differential Geom. 100 (2015), no. 2, 301–347. MR 3343834
Frederico Girão and Neilha M. Pinheiro, An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere, Ann. Global Anal. Geom. 52 (2017), no. 4, 413–424. MR 3735905, DOI 10.1007/s10455-017-9562-4
Pengfei Guan and Junfang Li, The quermassintegral inequalities for $k$-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. MR 2522433, DOI 10.1016/j.aim.2009.03.005
Pengfei Guan and Junfang Li, A mean curvature type flow in space forms, Int. Math. Res. Not. IMRN 13 (2015), 4716–4740. MR 3439091, DOI 10.1093/imrn/rnu081
Pengfei Guan and Junfang Li, A fully-nonlinear flow and quermassintegral inequalities (in Chinese), Sci. Sin. Math. 48 (2018), no. 1, 147–156, DOI 10.1360/N012017-00009.
Pengfei Guan, Junfang Li, and Mu Tao Wang, A volume preserving flow and the isoperimetric problem in warped product spaces, Trans. Am. Math. Soc. 372 (2019), 2777–2798, DOI 10.1090/tran/7661.
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Gerhard Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. MR 837523, DOI 10.1007/BF01388742
Gerhard Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math. 382 (1987), 35–48. MR 921165, DOI 10.1515/crll.1987.382.35
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Gerhard Huisken and Carlo Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45–70. MR 1719551, DOI 10.1007/BF02392946
Gerhard Huisken and Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009), no. 1, 137–221. MR 2461428, DOI 10.1007/s00222-008-0148-4
N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759, DOI 10.1007/978-94-010-9557-0
Matthias Makowski and Julian Scheuer, Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere, Asian J. Math. 20 (2016), no. 5, 869–892. MR 3622318, DOI 10.4310/AJM.2016.v20.n5.a2
James McCoy, The surface area preserving mean curvature flow, Asian J. Math. 7 (2003), no. 1, 7–30. MR 2015239, DOI 10.4310/AJM.2003.v7.n1.a2
James A. McCoy, Mixed volume preserving curvature flows, Calc. Var. Partial Differential Equations 24 (2005), no. 2, 131–154. MR 2164924, DOI 10.1007/s00526-004-0316-3
Julian Scheuer, Non-scale-invariant inverse curvature flows in hyperbolic space, Calc. Var. Partial Differential Equations 53 (2015), no. 1-2, 91–123. MR 3336314, DOI 10.1007/s00526-014-0742-9
Julian Scheuer, The inverse mean curvature flow in warped cylinders of non-positive radial curvature, Adv. Math. 306 (2017), 1130–1163. MR 3581327, DOI 10.1016/j.aim.2016.11.003
Julian Scheuer, Isotropic functions revisited, Arch. Math. (Basel) 110 (2018), no. 6, 591–604. MR 3803748, DOI 10.1007/s00013-018-1162-4
John I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), no. 3, 355–372. MR 1082861, DOI 10.1007/BF02571249
John I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), no. 1, 91–125. MR 1085136
Guofang Wang and Chao Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532–556. MR 3197666, DOI 10.1016/j.aim.2014.01.024
Yong Wei and Changwei Xiong, Inequalities of Alexandrov-Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere, Pacific J. Math. 277 (2015), no. 1, 219–239. MR 3393689, DOI 10.2140/pjm.2015.277.219
Chao Xia, A Minkowski type inequality in space forms, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 96, 8. MR 3523663, DOI 10.1007/s00526-016-1037-0