The Schläfli Formula in Einstein Manifolds with Boundary

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 En, et à la rigidité des variétés Ricci-plates à bord ombilique. Let M be a Riemannian (m + 1)-dimensional space-form of constant curvature K, and (Pt)t∈[0,1] a one-parameter family of polyhedra in M bounding compact domains, all having the same combinatorics. Call Vt the volume bounded by Pt, θi,t and Wi,t the dihedral angle and the (m− 1)-volume of the codimension 2 face i of Pt. The classical Schläfli formula (see [Mil94] or [Vin93]) is ∑ i Wi,t dθi,t dt = mK dVt 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 hypersurfaces 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.