Asymptotic convergence for a class of inverse mean curvature flows in ℝⁿ⁺¹

Chen, Li; Mao, Jing; Tu, Qiang; Wu, Di

doi:10.1090/proc/14686

Asymptotic convergence for a class of inverse mean curvature flows in $\mathbb {R}^{n+1}$
HTML articles powered by AMS MathViewer

by Li Chen, Jing Mao, Qiang Tu and Di Wu PDF

Proc. Amer. Math. Soc. 148 (2020), 379-392 Request permission

Abstract:

We consider star-shaped, strictly mean convex and closed hypersurfaces expanding by a class of inverse mean curvature flows in $\mathbb {R}^{n+1}$, and we prove that this evolution exists for all time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling.

References

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
Li Chen and Jing Mao, Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold, J. Geom. Anal. 28 (2018), no. 2, 921–949. MR 3790487, DOI 10.1007/s12220-017-9848-6
Qi Ding, The inverse mean curvature flow in rotationally symmetric spaces, Chinese Ann. Math. Ser. B 32 (2011), no. 1, 27–44. MR 2772224, DOI 10.1007/s11401-010-0626-z
Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995, DOI 10.1007/978-0-8176-8210-1
Yuxin Ge, Guofang Wang, Jie Wu, and Chao Xia, A Penrose inequality for graphs over Kottler space, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 755–782. MR 3311913, DOI 10.1007/s00526-014-0732-y
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
Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237–260. MR 3263518
Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. MR 2879249
Claus Gerhardt, Curvature flows in the sphere, J. Differential Geom. 100 (2015), no. 2, 301–347. MR 3343834
Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314. MR 1064876
Claus Gerhardt, Curvature problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR 2284727
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
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388. MR 3573332, DOI 10.1007/s11511-016-0140-6
G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices 20 (1997), 1045–1058. MR 1486695, DOI 10.1155/S1073792897000664
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
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
Haizhong Li and Yong Wei, On inverse mean curvature flow in Schwarzschild space and Kottler space, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 62, 21. MR 3639616, DOI 10.1007/s00526-017-1160-6
Haizhong Li, Xianfeng Wang, and Yong Wei, Surfaces expanding by non-concave curvature functions, Ann. Global Anal. Geom. 55 (2019), no. 2, 243–279. MR 3923539, DOI 10.1007/s10455-018-9625-1
Haizhong Li, Yong Wei, and Changwei Xiong, A geometric inequality on hypersurface in hyperbolic space, Adv. Math. 253 (2014), 152–162. MR 3148549, DOI 10.1016/j.aim.2013.12.003
Q. R. Li, W. M. Sheng, and X. J. Wang, Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., in press.
Q. R. Li, W. M. Sheng, and X. J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., in press.
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
S. Y. Lu, Inverse curvature flow in anti-de sitter-schwarzschild manifold, Commun. Anal. Geom. (to appear), arXiv:1609.09733, 2016.
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
Julian Scheuer, Gradient estimates for inverse curvature flows in hyperbolic space, Geom. Flows 1 (2015), no. 1, 11–16. MR 3338988, DOI 10.1515/geofl-2015-0002
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, Pinching and asymptotical roundness for inverse curvature flows in Euclidean space, J. Geom. Anal. 26 (2016), no. 3, 2265–2281. MR 3511477, DOI 10.1007/s12220-015-9627-1
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
J. Scheuer and X. Chao, Locally constrained inverse curvature flows, preprint, arXiv:1708.06125, 2017.
Julian Scheuer, Inverse curvature flows in Riemannian warped products, J. Funct. Anal. 276 (2019), no. 4, 1097–1144. MR 3906301, DOI 10.1016/j.jfa.2018.08.021
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
Hengyu Zhou, Inverse mean curvature flows in warped product manifolds, J. Geom. Anal. 28 (2018), no. 2, 1749–1772. MR 3790519, DOI 10.1007/s12220-017-9887-z