×

Surfaces expanding by non-concave curvature functions. (English) Zbl 1446.53077

The authors investigate the flow of convex surfaces in three-dimensional simply-connected space forms of curvature \(\kappa=0,\pm 1\). The flow is expandig by \(F^{-\alpha}\), here \(F\) is a smooth, symmetric and homogeneous of degree one function of the principal curvatures of the surface. The power \(\alpha\) satisfies \(0<\alpha \le 1\) for \(\kappa=0,-1\) and \(\alpha=1\) for \(\kappa=1\). The long-time existence and convergence of the flow is shown. An important ingredient is an estimate for the pinching ratio along the flow.

MSC:

53E10 Flows related to mean curvature
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J35 Heat and other parabolic equation methods for PDEs on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2), 151-171 (1994) · Zbl 0805.35048
[2] Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Diffe. Geom. 39(2), 407-431 (1994) · Zbl 0797.53044
[3] Andrews, B.: Fully nonlinear parabolic equations in two space variables, arXiv: math.DG/0402235 (2004)
[4] Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17-33 (2007) · Zbl 1129.53044
[5] Andrews, B.: Moving surfaces by non-concave curvature functions. Calc. Var. Partial Differ. Equ. 39(3-4), 649-657 (2010) · Zbl 1203.53062
[6] Andrews, B., Langford, M., McCoy, J.: Convexity estimates for surfaces moving by curvature functions. J. Differ. Geom. 99(1), 47-75 (2015) · Zbl 1310.53057
[7] Chow, B., Gulliver, R.: Aleksandrov reflection and nonlinear evolution equations. I. The \[n\] n-sphere and \[n\] n-ball. Calc. Var. Partial Differ. Equ. 4(3), 249-264 (1996) · Zbl 0851.58041
[8] Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299-314 (1990) · Zbl 0708.53045
[9] Gerhardt, C.: Closed Weingarten hypersurfaces in Riemannian manifolds. J. Differ. Geom. 43(3), 612-641 (1996) · Zbl 0861.53058
[10] Gerhardt, C.: Curvature Problems, Series in Geometry and Topology, vol. 39. International Press, Somerville (2006) · Zbl 1131.53001
[11] Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487-527 (2011) · Zbl 1252.53078
[12] Gerhardt, C.: Non-scale-invariant inverse curvature flows in Euclidean space. Calc. Var. Partial Differ. Equ. 49(1-2), 471-489 (2014) · Zbl 1284.53062
[13] Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geom. 100(2), 301-347 (2015) · Zbl 1395.53073
[14] Hamilton, R.: Convex hypersurfaces with pinched second fundamental form. Commun. Anal. Geom. 2, 167-172 (1994) · Zbl 0843.53002
[15] Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237-266 (1984) · Zbl 0556.53001
[16] Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems (Cetraro, 1996). In: Lecture Notes in Mathematics, vol. 1713, pp. 45-84. Springer, Berlin (1999) · Zbl 0942.35047
[17] Hung, P.-K., Wang, M.-T.: Inverse mean curvature flows in the hyperbolic 3-space revisited. Calc. Var. Partial Differ. Equ. 54(1), 119-126 (2015) · Zbl 1323.53076
[18] Kröner, H., Scheuer, J.: Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature, arXiv:1703.07087 · Zbl 1421.53068
[19] Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487-523, 670 (1982) (Russian) · Zbl 0511.35002
[20] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995) · Zbl 0816.35001
[21] Li, Q.-R.: Surfaces expanding by the power of the Gauss curvature flow. Proc. Am. Math. Soc. 138(11), 4089-4102 (2010) · Zbl 1204.53053
[22] Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996) · Zbl 0884.35001
[23] 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) · Zbl 1371.35101
[24] McCoy, J.A.: Curvature contraction flows in the sphere. Proc. Am. Math. Soc. 146(3), 1243-1256 (2018) · Zbl 1381.53122
[25] Neves, A.: Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds. J. Differ. Geom. 84(1), 191-229 (2010) · Zbl 1195.53096
[26] O’Neill, B.: Semi-Riemannian geometry. In: Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1983), With applications to relativity · Zbl 0531.53051
[27] Pipoli, G.: Inverse mean curvature flow in complex hyperbolic space, to appear on Annales scientifiques de l’ENS, arXiv:1610.01886 · Zbl 1391.53079
[28] Pipoli, G.: Inverse mean curvature flow in quaternionic hyperbolic space. Rendiconti Lincei Matematica e Applicazioni 29(1), 153-171 (2018) · Zbl 1391.53079
[29] Scheuer, J.: Gradient estimates for inverse curvature flows in hyperbolic space. Geom. Flows 1(1), 11-16 (2015) · Zbl 1348.35104
[30] Scheuer, J.: Non-scale-invariant inverse curvature flows in hyperbolic space. Calc. Var. Partial Differ. Equ. 53(1-2), 91-123 (2015) · Zbl 1317.53089
[31] Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130-1163 (2017) · Zbl 1357.53080
[32] Scheuer, J.: Inverse curvature flows in Riemannian warped products (2017), arxiv:1712.09521 · Zbl 1423.35109
[33] Schnürer, O.C.: Surfaces expanding by the inverse Gauß curvature flow. J. Reine Angew. Math. 600, 117-134 (2006) · Zbl 1119.53045
[34] Urbas, J.I.E.: On the expansion of star-shaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(1), 355-372 (1990) · Zbl 0691.35048
[35] Urbas, J.I.E.: An expansion of convex hypersurfaces. J. Differ. Geom. 33(1), 91-125 (1991) · Zbl 0746.53006
[36] Wei, Y.: New pinching estimates for Inverse curvature flows in space forms. J. Geom. Anal. (online first). https://doi.org/10.1007/s12220-018-0051-1 · Zbl 1416.53066
[37] Zhou, H.: Inverse mean curvature flows in warped product manifolds. J. Geom. Anal. 28(2), 1749-1772 (2018) · Zbl 1393.53069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.