Skip Navigation

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn068, 44 pages, doi:10.1093/imrn/rnn068 published on July 13, 2008
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Schlenker, J.-M.
Right arrow Articles by Souam, R.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Higher Schläfli Formulas and Applications. II: Vector-Valued Differential Relations

Jean-Marc Schlenker1 and Rabah Souam2

1 Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse Cedex 9, France
2 Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris Diderot - Paris 7, Géométrie et Dynamique, Site Chevaleret, Case 7012, 75205 - Paris Cedex 13, France

Correspondence: Correspondence to be sent to: schlenker{at}math.ups-tlse.fr

The classical Schläfli formula, and its "higher" analogs given in [23], are relations between the variations of the volumes and "curvatures" of faces of different dimensions of a polyhedra (which can be Euclidean, spherical, or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds.



References

  1. Alexandrov A. D. Convex Polyhedra (2005) Berlin: Springer. Translated from Russian by N. S. Dairbekov S. S. Kutateladze, and A. B. Sossinsky. Springer Monographs in Mathematics.
  2. Anderson Michael T. Formula curvature and volume renormalization of AHE metrics on 4-manifolds. Mathematical Research Letters (2001) 8(1–2):171–88.[ISI]
  3. Bezdek Károly, Connelly Robert. Pushing disks apart—the Kneser-Poulsen conjecture in the plane. Journal für die reine und angewandte Mathematik (2002) 553:221–36.[ISI]
  4. Bezdek Károly, Connelly Robert. The Kneser-Poulsen conjecture for spherical polytopes. Discrete and Computational Geometry (2004) 32(1):101–6.[CrossRef]
  5. Boileau Michel, Bernhard Leeb, Porti Joan. Geometrization of 3-dimensional orbifolds. Annals of Mathematics 2 (2005) 162(1):195–290.[ISI]
  6. Bonahon Francis. A Schläfli-type formula for convex cores of hyperbolic 3-manifolds. Journal of Differential Geometry (1998) 50(1):25–58.[ISI]
  7. Boileau Michel, Porti Joan. Geometrization of 3-orbifolds of cyclic type. Astérisque. 272 (2001): 208.
  8. Cheeger Jeff, Werner Müller, Schrader Robert. On the curvature of piecewise flat spaces. Communications in Mathematical Physics (1984) 92(3):405–54.[CrossRef][ISI]
  9. Connelly Robert. A counterexample to the rigidity conjecture for polyhedra. Institut des Hautes Etudes Scientifiques Publications Mathmatiques (1977) 47:333–38.[CrossRef]
  10. Csikós Balázs. A Schläfli-type formula for polytopes with curved faces and its application to the Kneser-Poulsen conjecture. Monatshefte für Mathematik (2006) 147(4):273–92.[CrossRef][ISI]
  11. Grosse-Brauckmann K., Korevaar N., Kusner R., Ratzkin J., Sullivan J. On nondegeneracy and regularity of the classifying map for coplanar cmc surfaces. forthcoming.
  12. Gluck Herman. Almost All Simply Connected Closed Surfaces are Rigid. Geometric Topology (1975) Berlin: Springer. 225–39. Lecture Notes in Mathematics 438.
  13. Korevaar N., Kusner R., Ratzkin J. On the nondegeneracy of constant mean curvature surfaces. Geometric and Functional Analysis (2006) 16(4):891–923.[CrossRef][ISI]
  14. Krasnov Kirill, Schlenker Jean-Marc. On the renormalized volume of hyperbolic 3-manifolds. Communications in Mathematical Physics (2008) 279:637–68. math.DG/0607081.[CrossRef][ISI]
  15. Kazdan Jerry L., Warner F. W. Curvature functions for compact 2-manifolds. Annals of Mathematics 2 (1974) 99:14–47.[CrossRef][ISI]
  16. Milnor J. The Schläfli Differential Equality. Collected Papers (1994) 1. Houston, TX: Publish or Perish.
  17. Regge T. General relativity without coordinates. Nuovo Cimento 10 (1961) 19:558–71.[CrossRef][ISI]
  18. Reilly R. C. Applications of the Hessian operator in a Riemannian manifold. Indiana University Mathematics Journal (1977) 26:459–72.[CrossRef][ISI]
  19. Rivin Igor, Hodgson Craig D. A characterization of compact convex polyhedra in hyperbolic 3-space. Inventiones Mathematicae (1993) 111:77–111.[CrossRef][ISI]
  20. Rivin Igor, Schlenker Jean-Marc. The Schläfli formula in Einstein manifolds with boundary. Electronic Research Announcements of the AMS (1999) 5:18–23.[CrossRef]
  21. Rosenberg Harold. Hypersurfaces of constant curvature in space forms. Bulletin des Sciences Mathmatiques (1993) 117(2):211–39.
  22. Schlenker Jean-Marc. Hyperbolic manifolds with convex boundary. Inventiones Mathematicae (2006) 163(1):109–69.[CrossRef][ISI]
  23. Schlenker Jean-Marc, Souam Rabah. Higher Schläfli formulas and applications. Compositio Mathematicae (2003) 135(1):1–24.[CrossRef]

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Schlenker, J.-M.
Right arrow Articles by Souam, R.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?