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ISSN 1079-6762

 

The Schläfli formula in Einstein manifolds with boundary


Authors: Igor Rivin and Jean-Marc Schlenker
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 18-23
MSC (1991): Primary 53C21; Secondary 53C25
Published electronically: March 22, 1999
MathSciNet review: 1669399
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Abstract: We give a smooth analogue of the classical Schläfli formula, relating the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersurfaces, and to the rigidity of Ricci-flat manifolds with umbilic boundaries.

RÉSUMÉ. On donne un analogue régulier de la formule classique de Schläfli, reliant la variation du volume borné par une hypersurface se déplaçant dans une variété d'Einstein à l'intégrale de la variation de la courbure moyenne. Puis nous l'étendons aux variations de la métrique à l'intérieur d'une variété d'Einstein riemannienne à bord. On l'applique aux cone-variétés d'Einstein, aux déformations isométriques d'hypersurfaces de $E^n$, et à la rigidité des variétés Ricci-plates à bord ombilique.


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Additional Information

Igor Rivin
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, G.B.
Email: irivin@ma.man.ac.uk

Jean-Marc Schlenker
Affiliation: Topologie et Dynamique (URA 1169 CNRS), Bât. 425, Université de Paris-Sud, 91405 Orsay Cedex, France
Email: jean-marc.schlenker@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S1079-6762-99-00057-8
PII: S 1079-6762(99)00057-8
Keywords: Vanishing theorems; null spaces
Received by editor(s): July 31, 1998
Published electronically: March 22, 1999
Communicated by: Walter Neumann
Article copyright: © Copyright 1999 American Mathematical Society