Abstract
In this paper, we first investigate the flow of convex surfaces in the space form \(\mathbb {R}^3(\kappa )~(\kappa =0,1,-1)\) expanding by \(F^{-\alpha }\), where F is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power \(\alpha \in (0,1]\) for \(\kappa =0,-1\) and \(\alpha =1\) for \(\kappa =1\). By deriving that the pinching ratio of the flow surface \(M_t\) is no greater than that of the initial surface \(M_0\), we prove the long time existence and the convergence of the flow. No concavity assumption of F is required. We also show that for the flow in \(\mathbb {H}^3\) with \(\alpha \in (0,1)\), the limit shape may not be necessarily round after rescaling.
Similar content being viewed by others
References
Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2), 151–171 (1994)
Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Diffe. Geom. 39(2), 407–431 (1994)
Andrews, B.: Fully nonlinear parabolic equations in two space variables, arXiv: math.DG/0402235 (2004)
Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)
Andrews, B.: Moving surfaces by non-concave curvature functions. Calc. Var. Partial Differ. Equ. 39(3–4), 649–657 (2010)
Andrews, B., Langford, M., McCoy, J.: Convexity estimates for surfaces moving by curvature functions. J. Differ. Geom. 99(1), 47–75 (2015)
Chow, B., Gulliver, R.: Aleksandrov reflection and nonlinear evolution equations. I. The \(n\)-sphere and \(n\)-ball. Calc. Var. Partial Differ. Equ. 4(3), 249–264 (1996)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)
Gerhardt, C.: Closed Weingarten hypersurfaces in Riemannian manifolds. J. Differ. Geom. 43(3), 612–641 (1996)
Gerhardt, C.: Curvature Problems, Series in Geometry and Topology, vol. 39. International Press, Somerville (2006)
Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487–527 (2011)
Gerhardt, C.: Non-scale-invariant inverse curvature flows in Euclidean space. Calc. Var. Partial Differ. Equ. 49(1–2), 471–489 (2014)
Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geom. 100(2), 301–347 (2015)
Hamilton, R.: Convex hypersurfaces with pinched second fundamental form. Commun. Anal. Geom. 2, 167–172 (1994)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
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)
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)
Kröner, H., Scheuer, J.: Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature, arXiv:1703.07087
Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487–523, 670 (1982) (Russian)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995)
Li, Q.-R.: Surfaces expanding by the power of the Gauss curvature flow. Proc. Am. Math. Soc. 138(11), 4089–4102 (2010)
Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)
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)
McCoy, J.A.: Curvature contraction flows in the sphere. Proc. Am. Math. Soc. 146(3), 1243–1256 (2018)
Neves, A.: Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds. J. Differ. Geom. 84(1), 191–229 (2010)
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
Pipoli, G.: Inverse mean curvature flow in complex hyperbolic space, to appear on Annales scientifiques de l’ENS, arXiv:1610.01886
Pipoli, G.: Inverse mean curvature flow in quaternionic hyperbolic space. Rendiconti Lincei Matematica e Applicazioni 29(1), 153–171 (2018)
Scheuer, J.: Gradient estimates for inverse curvature flows in hyperbolic space. Geom. Flows 1(1), 11–16 (2015)
Scheuer, J.: Non-scale-invariant inverse curvature flows in hyperbolic space. Calc. Var. Partial Differ. Equ. 53(1–2), 91–123 (2015)
Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)
Scheuer, J.: Inverse curvature flows in Riemannian warped products (2017), arxiv:1712.09521
Schnürer, O.C.: Surfaces expanding by the inverse Gauß curvature flow. J. Reine Angew. Math. 600, 117–134 (2006)
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)
Urbas, J.I.E.: An expansion of convex hypersurfaces. J. Differ. Geom. 33(1), 91–125 (1991)
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
Zhou, H.: Inverse mean curvature flows in warped product manifolds. J. Geom. Anal. 28(2), 1749–1772 (2018)
Acknowledgements
The first author was supported in part by NSFC Grant No. 11671214. The second author was supported in part by NSFC Grant No. 11571185 and the Fundamental Research Funds for the Central Universities. The third author was supported by Ben Andrews throughout his Australian Laureate Fellowship FL150100126 of the Australian Research Council. The authors would like to thank Ben Andrews and Julian Scheuer for comments on the earlier version of this paper and the referees for carefully reading this paper and providing many helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, H., Wang, X. & Wei, Y. Surfaces expanding by non-concave curvature functions. Ann Glob Anal Geom 55, 243–279 (2019). https://doi.org/10.1007/s10455-018-9625-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-018-9625-1