ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 18–23 (March 22, 1999) S 1079-6762(99)00057-8 THE SCHL ¨ AFLI FORMULA IN EINSTEIN MANIFOLDS WITH BOUNDARY IGOR RIVIN AND JEAN-MARC SCHLENKER (Communicated by Walter Neumann) Abstract. We give a smooth analogue of the classical Schl¨ afli 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. R´ esum´ e. On donne un analogue r´ egulier de la formule classique de Schl¨afli, reliant la variation du volume born´ e par une hypersurface se d´ epla¸cant dans une vari´ et´ e d’Einstein `a l’int´ egrale de la variation de la courbure moyenne. Puis nous l’´ etendons aux variations de la m´ etrique ` a l’int´ erieur d’une vari´ et´ e d’Einstein riemannienne `a bord. On l’applique aux cone-vari´ et´ es d’Einstein, aux d´ eformations isom´ etriques d’hypersurfaces de E n , et `a la rigidit´ e des vari´ et´ es Ricci-plates `a bord ombilique. Let M be a Riemannian (m + 1)-dimensional space-form of constant curvature K, and (P t ) t∈[0,1] a one-parameter family of polyhedra in M bounding compact domains, all having the same combinatorics. Call V t the volume bounded by P t , θ i,t and W i,t the dihedral angle and the (m − 1)-volume of the codimension 2 face i of P t . The classical Schl¨afli formula (see [Mil94] or [Vin93]) is  i W i,t dθ i,t dt = mK dV t dt . (1) This formula has been extended and used on several occasions recently; see for instance [Hod86], [Bon]. We give a smooth version of this formula, for 1-parameter families of hyper- surfaces in (Riemannian of Lorentzian) Einstein manifolds. Then we extend it to variations of an Einstein metric inside a manifold with boundary (a much more general process in dimension above 3). Finally, we give three applications: to the variation of the volume of Einstein cone-manifolds, to isometric deformations of hypersurfaces in the Euclidean space, and to the rigidity of Ricci-flat manifolds with umbilic boundaries. The reader can find the details in [RS98]. Throughout this paper, M is an Einstein manifold of dimension m +1 ≥ 3, and D is its Levi-Civita connection. When dealing with a hypersurface Σ (resp. with Received by the editors July 31, 1998. 1991 Mathematics Subject Classification. Primary 53C21; Secondary 53C25. Key words and phrases. Vanishing theorems; null spaces. c 1999 American Mathematical Society 18