Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T09:11:50.140Z Has data issue: false hasContentIssue false
  • English
  • Français

On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes

Part of: General relativity Spaces and manifolds of mappings Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

Klaus Ecker
Klaus Ecker
Affiliation:
Department of Mathematics, The University of Melbourne, Parkville, Victoria 3052, Australia, email: [email protected]
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.

We prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.

MSC classification

Secondary: 58D25: Equations in function spaces; evolution equations 83C99: None of the above, but in this section 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 1 , August 1993 , pp. 41 - 59
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bartnik, R., ‘Existence of maximal surfaces in asymptotically flat spacetimes’, Comm. Math. Phys. 94 (1984), 155–175.CrossRefGoogle Scholar
[2]Bartnik, R., ‘Regularity of variational maximal surfaces’, Acta Math. 161 (1988), 145–181.CrossRefGoogle Scholar
[3]Bartnik, R., Chruściel, P. T. and Murchadha, N.Ō., ‘On maximal surfaces in asymptotically flat space-times’, Comm. Math. Phys. 130 (1990), 95–109.CrossRefGoogle Scholar
[4]Bishop, R. and Crittenden, R., Geometry of manifolds (Academic Press, New York, 1964).Google Scholar
[5]Cheng, S. Y. and Yau, S. T., ‘Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces’, Ann. of Math. 104 (1976), 407–419.CrossRefGoogle Scholar
[6]Ecker, K. and Huisken, G., ‘Mean curvature evolution of entire graphs’, Ann. of Math. 130 (1989), 453–471.CrossRefGoogle Scholar
[7]Ecker, K. and Huisken, G., ‘Interior estimates for hypersurfaces moving by mean curvature’, Invent. Math. 105 (1991), 547–569.CrossRefGoogle Scholar
[8]Ecker, K. and Huisken, G., ‘Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes’, Comm. Math. Phys. 135 (1991), 595–613.CrossRefGoogle Scholar
[9]Gerhardt, C., ‘H-surfaces in Lorentzian manifolds’, Comm. Math. Phys. 89 (1983), 523–553.CrossRefGoogle Scholar
[10]Hawking, S. and Ellis, G., The large scale structure of spacetime (Cambridge Univ. Press, London, 1973).CrossRefGoogle Scholar
[11]Huisken, G., ‘Flow by mean curvature of convex surfaces into spheres’, J. Differential Geom. 20 (1984), 237–266.CrossRefGoogle Scholar
[12]Huisken, G., ‘Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature’, Invent. Math. 84 (1986), 463–480.CrossRefGoogle Scholar
[13]Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs (Amer. Math. Soc., Providence), 1968).CrossRefGoogle Scholar
[14]Liao, G. G. and Tam, L. F., ‘On the heat equation for harmonic maps for non-compact manifolds’, Pacific J. Math. 153 (1991), 129–145.CrossRefGoogle Scholar
[15]Marsden, J. E. and Tipler, F. J., ‘Maximal hypersurfaces and foliations of constant mean curvature in general relativity’, Phys. Rep. 66 (3) (1980), 109–139.CrossRefGoogle Scholar
[16]Stavrou, N., Flow of spacelike hypersurfaces in Minkowski space (Honours Thesis, University of Melbourne, 1991).Google Scholar