Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T03:07:21.646Z Has data issue: false hasContentIssue false
  • English
  • Français

Contracting spacelike hypersurfaces by their inverse mean curvature

Part of: Global differential geometry General relativity

Published online by Cambridge University Press:  09 April 2009

Michael Holder
Michael Holder
Affiliation:
Mathematisches Institut Universität TübingenAuf der Morgenstelle 10 D-72076 TübingenGermany e-mail: michael@moebius.mathematik.uni-tuebingen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work we study the behaviour of compact, smooth, orientable, spacelike hypersurfaces without boundary, which are immersed in cosmological spacetimes and move under the inverse mean curvature flow. We prove longtime existence and regularity of a solution to the corresponding nonlinear parabolic system of partial differential equations.

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53C50: Lorentz manifolds, manifolds with indefinite metrics 83C30: Asymptotic procedures (radiation, news functions, $scr H$-spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 285 - 300
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Bartnik, R., ‘Existence of maximal surfaces in asymptotically flat spacetimes’, Comm. Math. Phys. 94 (1984), 155–175.CrossRefGoogle Scholar
[2]Bartnik, R., ‘Remarks on cosmological spacetimes and constant mean curvature surfaces’, Comm. Math. Phys. 117 (1988), 615–624.CrossRefGoogle Scholar
[3]Cheng, S.-Y. and Yau, S.-T., ‘Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces’, Ann. of Math. 104 (1976), 405–419.CrossRefGoogle Scholar
[4]Choquet-Bruhat, Y., ‘Maximal submanifolds and submanifolds with constant mean extrinsic curvature of a Lorentzian manifold’, Ann. Scuola Norm. Sup. Pisa 3 (1976), 361–376.Google Scholar
[5]Ecker, K. and Huisken, G., ‘Interior estimates for hypersurfaces moving by mean curvature’, Invent. Math. 105 (1991), 547–569.CrossRefGoogle Scholar
[6]Ecker, K. and Huisken, G., ‘Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes’, Comm. Math. Phys. 35 (1991), 595–613.CrossRefGoogle Scholar
[7]Fischer, A. and Marsden, J., ‘Topics in the dynamics of general relativity, in isolated gravitating systems in general relativity’, Italian Phys. Soc. (1979), 322–395.Google Scholar
[8]Gerhardt, C., ‘H-surfaces in Lorentzian manifolds’, Comm. Math. Phys. 89 (1983), 523–553.CrossRefGoogle Scholar
[9]Gerhardt, C., ‘Flow of nonconvex hypersurfaces into spheres’, J. Differential Geom. 32 (1990), 299–314.CrossRefGoogle Scholar
[10]Hamilton, R. S., ‘Three-manifolds with positive Ricci curvature’, J. Differential Geom. 17 (1982), 255–306.CrossRefGoogle Scholar
[11]Hamilton, R. S., ‘Four-manifolds with positive curvature operator’, J. Differential Geom. 24 (1986), 153–179.CrossRefGoogle Scholar
[12]Hamilton, R. S., Lecture notes (Hawaii, 1989).Google Scholar
[13]Hawking, S. W. and Ellis, G. F. R., The large scale structure of space-time, Cambridge Monographs Math. Phys. (Cambridge University Press, Cambridge, 1973).CrossRefGoogle Scholar
[14]Huisken, G., ‘Flow by mean curvature of convex surfaces into spheres’, J. Differential Geom. 10 (1984), 237–266.Google Scholar
[15]Huisken, G., ‘Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature’, Invent. Math. 84 (1986), 463–480.CrossRefGoogle Scholar
[16]Huisken, G., ‘Deforming hypersurfaces of the sphere by their mean curvature’, Math. Z. 195 (1987), 205–219.CrossRefGoogle Scholar
[17]Huisken, G. and Ilmanen, T., ‘The Riemannian Penrose inequality’, Internat. Math. Res. Notices 20 (1997), 1045–1058.CrossRefGoogle Scholar
[18]Krylov, N. V., Nonlinear elliptic and parabolic equations of the second order (D. Reidel Publ. Co., Dordrecht, 1987).CrossRefGoogle Scholar
[19]Ladyshenzkaya, O. A., Solonnikov, V. A. and Ural'ceva, N. N., Linear and quasilinear equations of parabolic type, English translation: Transl. Math. Monographs, Vol. 23 (Amer. Math. Soc., Rhode Island, 1967).Google Scholar
[20]Nishikawa, S., ‘On maximal space-like hypersurfaces in a Lorentzian manifold’, Nagoya Math. J. 95 (1984), 117–124.CrossRefGoogle Scholar
[21]O'Neill, B., Semi-Riemannian geometry with applications to relativity, Pure and Appl. Math. 103 (Academic Press, New York, 1983).Google Scholar
[22]Schoen, R.Simon, L. and Yau, S.-T., ‘Curvature estimates for minimal hypersurfaces’, Acta Math. 134 (1975), 275–288.CrossRefGoogle Scholar
[23]Urbas, J. I. E., ‘On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures’, Math. Z. 205 (1990), 355–372.CrossRefGoogle Scholar