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We give a smooth analogue of the classical Schl{\"a}fli 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. \\ \\ \noindent {\sc R\'esum\'e.} On donne un analogue r{\'e}gulier de la formule classique de Schl{\"a}fli, reliant la variation du volume born{\'e} par une hypersurface se d{\'e}pla{\c c}ant dans une vari{\'e}t{\'e} d'Einstein {\`a} l'int{\'e}grale de la variation de la courbure moyenne. Puis nous l'{\'e}tendons aux variations de la m{\'e}trique {\`a} l'int{\'e}rieur d'une vari{\'e}t{\'e} d'Einstein riemannienne {\`a} bord. On l'applique aux cone-vari{\'e}t{\'e}s d'Einstein, aux d{\'e}formations isom{\'e}triques d'hypersurfaces de $E^n$, et {\`a} la rigidit{\'e} des vari{\'e}t{\'e}s Ricci-plates {\`a} bord ombilique.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 18-23
• Publisher Identifier: S 1079-6762(99)00057-8
• 1991 Mathematics Subject Classification. Primary 53C21
• Key words and phrases. Vanishing theorems; null spaces
• Received by the editors July 31, 1998
• Posted on March 22, 1999
• Communicated by Walter Neumann
• Comments (When Available)

Igor Rivin

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, G.B.

E-mail address: irivin@ma.man.ac.uk

Jean-Marc Schlenker

Topologie et Dynamique (URA 1169 CNRS), B{\^a}t. 425, Uni\-ver\-sit{\'e} de Paris-Sud, 91405 Orsay Cedex, France

E-mail address: jean-marc.schlenker@math.u-psud.fr