Geodesic
The Schläfli formula in Einstein manifolds with boundary
Rivin, Igor1 ; Schlenker, Jean-Marc2
1 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, G.B.
2 Topologie et Dynamique (URA 1169 CNRS), Bât. 425, Université de Paris-Sud, 91405 Orsay Cedex, France
Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 18-23.

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S1079-6762-99-00057-8

Rivin, Igor 1 ; Schlenker, Jean-Marc 2

1 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, G.B.
2 Topologie et Dynamique (URA 1169 CNRS), Bât. 425, Université de Paris-Sud, 91405 Orsay Cedex, France 
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Rivin, Igor; Schlenker, Jean-Marc. The Schläfli formula in Einstein manifolds with boundary. Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 18-23. doi : 10.1090/S1079-6762-99-00057-8. http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-99-00057-8/

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