Inverse curvature flows of rotation hypersurfaces. (English) Zbl 1484.53124
Summary: We consider the inverse curvature flows of smooth, closed and strictly convex rotation hypersurfaces in space forms \(\mathbb{M}_k^{n+1}\) with speed function given by \(F^{-\alpha}\), where \(\alpha \in (0, 1]\) for \(\kappa = 0, - 1, \alpha =1\) for \(\kappa = 1\) and \(F\) is a smooth, symmetric, strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces. We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value, and prove the long time existence and convergence of the flows. No second derivatives conditions are required on \(F\).
MSC:
| 53E99 | Geometric evolution equations |
| 53E10 | Flows related to mean curvature |
| 53B25 | Local submanifolds |
References:
| [1] | Andrews, B., Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom., 39, 2, 407-431 (1994) · Zbl 0797.53044 · doi:10.4310/jdg/1214454878 |
| [2] | Andrews, B.: Fully nonlinear parabolic equations in two space variables, arXiv: math.DG/0402235 (2004) |
| [3] | Andrews, B., Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608, 17-33 (2007) · Zbl 1129.53044 |
| [4] | Andrews, B.; Huang, Z.; Li, H., Uniqueness for a class of embedded Weingarten hypersurfaces in S’^n+^1, Introduction to Modern Mathematics, 95-107 (2015), Somerville, MA: Int. Press, Somerville, MA · Zbl 1359.53050 |
| [5] | Chow, B.; Gulliver, R., Aleksandrov reflection and nonlinear evolution equations. I. The n-sphere and n-ball, Calc. Var. Partial Differential Equations, 4, 3, 249-264 (1996) · Zbl 0851.58041 · doi:10.1007/BF01254346 |
| [6] | do Carmo, M.; Dajczer, M., Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc., 277, 2, 685-709 (1983) · Zbl 0518.53059 · doi:10.1090/S0002-9947-1983-0694383-X |
| [7] | Gerhardt, G., Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32, 1, 299-314 (1990) · Zbl 0708.53045 · doi:10.4310/jdg/1214445048 |
| [8] | Gerhardt, G., Curvature Problems, Series in Geometry and Topology (2006), Somerville, MA: International Press, Somerville, MA · Zbl 1131.53001 |
| [9] | Gerhardt, G., Inverse curvature flows in hyperbolic space, J. Differential Geom., 89, 3, 487-527 (2011) · Zbl 1252.53078 · doi:10.4310/jdg/1335207376 |
| [10] | Gerhardt, G., Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differential Equations, 49, 1-2, 471-489 (2014) · Zbl 1284.53062 · doi:10.1007/s00526-012-0589-x |
| [11] | Gerhardt, G., Curvature flows in the sphere, J. Differential Geom., 100, 2, 301-347 (2015) · Zbl 1395.53073 · doi:10.4310/jdg/1430744123 |
| [12] | Guan, P.; Li, J., The quermassintegral inequalities for k-convex starshaped domains, Adv. Math., 221, 1725-1732 (2009) · Zbl 1170.53058 · doi:10.1016/j.aim.2009.03.005 |
| [13] | Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20, 1, 237-266 (1984) · Zbl 0556.53001 · doi:10.4310/jdg/1214438998 |
| [14] | Huisken, G.; Ilmanen, T., The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59, 353-438 (2001) · Zbl 1055.53052 · doi:10.4310/jdg/1090349447 |
| [15] | Kröner, H.; Scheuer, J., Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature, Math. Nachr., 292, 7, 1514-1529 (2019) · Zbl 1421.53068 · doi:10.1002/mana.201700370 |
| [16] | Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 46, 3, 487-523 (1982) · Zbl 0511.35002 |
| [17] | Li, Q. R., Surfaces expanding by the power of the Gauss curvature flow, Proc. Amer. Math. Soc., 138, 11, 4089-4102 (2010) · Zbl 1204.53053 · doi:10.1090/S0002-9939-2010-10431-8 |
| [18] | Li, H.; Wang, X.; Wei, Y., Surfaces expanding by non-concave curvature functions, Ann. Global Anal. Geom., 55, 2, 243-279 (2019) · Zbl 1446.53077 · doi:10.1007/s10455-018-9625-1 |
| [19] | Lieberman, G. M., Second Order Parabolic Differential Equations (1996), Singapore: World Scientific, Singapore · Zbl 0884.35001 · doi:10.1142/3302 |
| [20] | Makowski, M.; Scheuer, J., Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere, Asian J. Math., 20, 5, 869-892 (2017) · Zbl 1371.35101 · doi:10.4310/AJM.2016.v20.n5.a2 |
| [21] | McCoy, J. A.; Mofarreh, F. Y Y.; Wheeler, V. M., Fully nonlinear curvature flow of axially symmetric hypersurfaces, Nonlinear Differ. Equ. Appl., 22, 325-343 (2015) · Zbl 1400.53057 · doi:10.1007/s00030-014-0287-9 |
| [22] | Scheuer, J., Gradient estimates for inverse curvature flows in hyperbolic space, Geometric Flows, 1, 1, 11-16 (2015) · Zbl 1348.35104 · doi:10.1515/geofl-2015-0002 |
| [23] | Scheuer, J., Non-scale-invariant inverse curvature flows in hyperbolic space, Calc. Var. Partial Differential Equations, 53, 1-2, 91-123 (2015) · Zbl 1317.53089 · doi:10.1007/s00526-014-0742-9 |
| [24] | Schnürer, O. C., Surfaces expanding by the inverse Gauß curvature flow, J. Reine Angew. Math., 600, 117-134 (2006) · Zbl 1119.53045 |
| [25] | Urbas, J. I E., On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205, 355-372 (1990) · Zbl 0691.35048 · doi:10.1007/BF02571249 |
| [26] | Urbas, J. I E., An expansion of convex hypersurfaces, J. Differential Geom., 33, 1, 91-125 (1991) · Zbl 0746.53006 · doi:10.4310/jdg/1214446031 |
| [27] | Wei, Y., New pinching estimates for inverse curvature flows in space forms, J. Geom. Anal., 29, 2, 1555-1570 (2019) · Zbl 1416.53066 · doi:10.1007/s12220-018-0051-1 |
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