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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]
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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

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