Remote Access Electronic Research Announcements

Electronic Research Announcements

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AR97] F. O. Almgren and I. Rivin. The mean curvature integral is invariant under bending. Manuscript, 1997.
  • [BG93] Marcel Berger and Bernard Gostiaux, Géométrie différentielle: variétés, courbes et surfaces, 2nd ed., Mathématiques. [Mathematics], Presses Universitaires de France, Paris, 1992 (French, with French summary). MR 1207362
  • [Biq97] O. Biquard. Métriques d'Einstein asymptotiquement symétriques. Prépublication no. 97-24, Ecole Polytechnique, 1997.
  • [Ble96] David D. Bleecker, Volume increasing isometric deformations of convex polyhedra, J. Differential Geom. 43 (1996), no. 3, 505–526. MR 1412676
  • [Bon] F. Bonahon. A Schläfli-type formula for convex cores of hyperbolic 3-manifolds. Journal of Diff. Geometry. To appear.
  • [Con77] Robert Connelly, A counterexample to the rigidity conjecture for polyhedra, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 333–338. MR 0488071
  • [DeT81] Dennis M. DeTurck, Existence of metrics with prescribed Ricci curvature: local theory, Invent. Math. 65 (1981/82), no. 1, 179–207. MR 636886, 10.1007/BF01389010
  • [GL91] C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186–225. MR 1112625, 10.1016/0001-8708(91)90071-E
  • [Her79] Gustav Herglotz, Gesammelte Schriften, Vandenhoeck & Ruprecht, Göttingen, 1979 (German). With introductory articles by Peter Bergmann, S. S. Chern, Ronald B. Guenther, Claus Müller, Theodor Schneider and H. Wittich; Edited and with a foreword by Hans Schwerdtfeger. MR 526569
  • [Hod86] C. Hodgson. PhD Thesis, Princeton University, 1986.
  • [Lie00] H. Liebmann. Ueber die Verbiegung der geschlossenen Fläschen positiver Krümmung. Math. Annalen, 53:81-112, 1900.
  • [Mil94] John Milnor, Collected papers. Vol. 1, Publish or Perish, Inc., Houston, TX, 1994. Geometry. MR 1277810
  • [Nir53] Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. MR 0058265
  • [Pog73] A. V. Pogorelov, Extrinsic geometry of convex surfaces, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 35. MR 0346714
  • [RS98] I. Rivin and J-M. Schlenker. Schläfli formula for Einstein manifolds. IHES Preprint, 1998.
  • [San76] Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364
  • [Sch98] J.-M. Schlenker. Einstein manifolds with convex boundaries. Prépublication no. 98-12, Université de Paris-Sud, 1998.
  • [SP97] E. Suarez-Peiró. A Schläfli formula for simplices in semi-Riemannian hyperquadrics, Gauss-Bonnet formulas for simplices in the de Sitter sphere and the dual volume of a hyperbolic simplex. Manuscript, 1997.
  • [Spi75] Michael Spivak, A comprehensive introduction to differential geometry. Vol. One, Published by M. Spivak, Brandeis Univ., Waltham, Mass., 1970. MR 0267467
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, Published by M. Spivak, Brandeis Univ., Waltham, Mass., 1970. MR 0271845
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, Publish or Perish, Inc., Boston, Mass., 1975. MR 0372756
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394452
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394453
  • [Vin93] D. V. Alekseevskij, È. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 1–138. MR 1254932, 10.1007/978-3-662-02901-5_1

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 53C21, 53C25

Retrieve articles in all journals with MSC (1991): 53C21, 53C25


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
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