Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-06T09:57:29.565Z Has data issue: false hasContentIssue false
  • English
  • Français

On smooth time functions

Published online by Cambridge University Press:  01 November 2011

ALBERT FATHI  and
ANTONIO SICONOLFI
ALBERT FATHI
Affiliation:
Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon, France. e-mail: [email protected]
ANTONIO SICONOLFI
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale A. Moro, 2, 00185 Roma, Italy. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.

Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 2 , March 2012 , pp. 303 - 339
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aubin, J. P. and Cellina, A.Differential Inclusion. Grundlehren der mathematischen Wissenschaften 264 (Springer–Verlag, 1984).CrossRefGoogle Scholar
[2]Beem, J. K., Ehrlich, P. E. and Easley, K. L.Global Lorentzian geometry. Monograph and Textbooks in Pure Appl. Math. 202 (Marcel Dekker Inc., 1996).Google Scholar
[3]Beer, G.Topologies on Closed and Closed Convex Sets (Kluwer Academic Publishers, 1993).CrossRefGoogle Scholar
[4]Bernal, A. N and Sánchez, M.On smooth Cauchy hypersurfaces and Geroch's splitting theorem. Comm. Math Phys. 243 (2003), 461–470.CrossRefGoogle Scholar
[5]Dattorro, J.Convex Optimization and Euclidean Distance Geometry (Meboo Publishing, USA), 2005.Google Scholar
[6]Fathi, A. and Maderna, E.Weak KAM theorem on non compact manifolds Nonlinear Diff. Eq. Appl. 14 (2007), 1–27.CrossRefGoogle Scholar
[7]Fathi, A. and Siconolfi, A.PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations 22 (2005), 185–228.CrossRefGoogle Scholar
[8]Fathi, A.Weak KAM Theorem in Lagrangian Dynamics (Cambridge University Press), to appear.Google Scholar
[9]Geroch, R.Domain of dependence. J. Math. Phys. 11 (1970), 437–449.CrossRefGoogle Scholar
[10]Hawking, S. W. and Ellis, G.F.RThe large scale structure of space-time. Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1973).CrossRefGoogle Scholar
[11]O'Neill, B.Semi–Riemannian Geometry (Academic Press, 1983).Google Scholar
[12]Siconolfi, A. and Terrone, G.A metric approach to the converse lyapunov theorem for continuous multivalued dynamics. Nonlinearity, 20 (2007), 1077–1093.CrossRefGoogle Scholar