SpringerLink
Log in

Fuchsian convex bodies: basics of Brunn–Minkowski theory

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

The hyperbolic space \({\mathbb{H}^d}\) can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that \({\mathbb{H}^d/\Gamma}\) is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A.D. Alexandrov. On the theory of mixed volumes II. Mat Sbornik, 44 (1937), 1205–1238. (Russian. Translated in [Ale96]).

    Google Scholar 

  2. A.D. Alexandrov. On the theory of mixed volumes IV. Mat Sbornik, (2)3 (1938), 227–249. (Russian. Translated in [Ale96]).

  3. A.D. Alexandrov. Selected Works. Part I. Classics of Soviet Mathematics, Vol. 4. Gordon and Breach Publishers, Amsterdam. Selected Scientific Papers, Translated from the Russian by P.S.V. Naidu, Edited and with a preface by Yu.G. Reshetnyak and S.S. Kutateladze (1996).

  4. A.D. Alexandrov. Convex Polyhedra. Springer Monographs in Mathematics. Springer-Verlag, Berlin. Translated from the 1950 Russian edition by N.S. Dairbekov, S.S. Kutateladze and A.B. Sossinsky, With comments and bibliography by V.A. Zalgaller and appendices by L.A. Shor and Yu.A. Volkov (2005).

  5. L. Andersson, T. Barbot, R. Benedetti, F. Bonsante, W.M. Goldman, F. Labourie, K.P. Scannell, and J.-M. Schlenker. Notes on: “Lorentz spacetimes of constant curvature” [Geometriae Dedicata 126 (2007), 3–45; mr2328921] by G. Mess. Geometriae Dedicata, 126 (2007) 47–70.

  6. Bahn H., Ehrlich P.: A Brunn-Minkowski type theorem on the Minkowski spacetime. Canadian Journal of Mathematics 51(3), 449–469 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbot T.: Globally hyperbolic flat space-times. Journal of Geometry and Physics 53(2), 123–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barbot T., Béguin F., Zeghib A.: Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: Application to the Minkowski problem in the Minkowski space. Ann Inst Fourier (Grenoble) 61(2), 511–591 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Barbot and A. Zeghib. Group actions on Lorentz spaces, mathematical aspects: A survey. In: The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel (2004), pp. 401–439.

  10. Benedetti R., Petronio C.: Lectures on Hyperbolic Geometry. Universitext. Springer-Verlag, Berlin (1992)

    Book  MATH  Google Scholar 

  11. J. Bertrand. Prescription of Gauss Curvature on compact hyperbolic orbifolds, new version of the 2010 preprint Prescription of Gauss Curvature Using Optimal Mass Transport (2012).

  12. T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, ID. Translated from the German and edited by L. Boron, C. Christenson and B. Smith (1987).

  13. Bonsante F.: Flat spacetimes with compact hyperbolic Cauchy surfaces. Journal of Differential Geometry 69(3), 441–521 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Y.D. Burago and V.A. Zalgaller. Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 285. Springer-Verlag, Berlin. Translated from the Russian by A.B. Sosinskiĭ. Springer Series in Soviet Mathematics (1988).

  15. H. Busemann. Convex Surfaces. Dover, Mineola. Reprint of the 1958 original (2008).

  16. Charney R., Davis M., Moussong G.: Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds. Michigan Mathematical Journal 44(1), 201–208 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cheng S.Y., Yau S.T.: On the regularity of the solution of the n-dimensional Minkowski problem. Communications on Pure and Applied Mathematics 29(5), 495–516 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Epstein D.B.A., Penner R.C.: Euclidean decompositions of noncompact hyperbolic manifolds. Journal of Differential Geometry 27(1), 67–80 (1988)

    MathSciNet  MATH  Google Scholar 

  19. Espinar J.M., Gálvez J.A., Mira P.: Hypersurfaces in \({\mathbb{H}^{n+1}}\) and conformally invariant equations: The generalized Christoffel and Nirenberg problems. Journal of The European Mathematical Society 11(4), 903–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fillastre F.: Fuchsian polyhedra in Lorentzian space-forms. Annals of Mathematics 350(2), 417–453 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Fillastre. Polygons of the Lorentzian plane and spherical simplexes (2013).

  22. F. Fillastre and G. Veronelli. Lorentzian area measures and the Christoffel problem. In preparation (2012).

  23. Gardner R.J.: The Brunn–Minkowski inequality. Bulletin of the American Mathematical Society (N.S.) 39(3), 355–405 (2002)

    Article  MATH  Google Scholar 

  24. M. Gromov. Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 9. Springer-Verlag, Berlin (1986).

  25. Gromov M., Piatetski-Shapiro I.: Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. 66, 93–103 (1988)

    MathSciNet  MATH  Google Scholar 

  26. P. Guan, X.-N. Ma, N. Trudinger, and X. Zhu. A form of Alexandrov-Fenchel inequality. Pure and Applied Mathematics Quarterly, (4, Special Issue: In honor of Joseph J. Kohn. Part 2)6 (2010), 999–1012.

  27. Hamilton R.S.: The inverse function theorem of Nash and Moser. Bulletin of the American Mathematical Society (N.S.) 7(1), 65–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Hörmander. Notions of Convexity. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA. Reprint of the 1994 edition (2007).

  29. I. Iskhakov. On hyperbolic Surface Tessalations and Equivariant Spacelike Convex Polyhedral Surfaces in Minkowski Space. PhD thesis, Ohio State University (2000).

  30. T. Kato. Perturbation Theory for Linear Operators. Classics in Mathematics. Springer-Verlag, Berlin. Reprint of the 1980 edition (1995).

  31. Katok S.: Fuchsian Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992)

    MATH  Google Scholar 

  32. Labourie F., Schlenker J.-M.: Surfaces convexes fuchsiennes dans les espaces lorentziens à à courbure constante. Mathematische Annalen 316(3), 465–483 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Langevin, G. Levitt, and H. Rosenberg. Hérissons et multihérissons (enveloppes parametrées par leur application de Gauss). In: Singularities (Warsaw, 1985). Banach Center Publ., Vol. 20. PWN, Warsaw (1988), pp. 245–253.

  34. K. Leichtweiß. Convexity and differential geometry. In: Handbook of Convex Geometry, Vol. A, B. North-Holland, Amsterdam (1993), pp. 1045–1080.

  35. Lopesde Lima L., Soaresde Lira J.H.: The Christoffel problem in Lorentzian geometry. Journal of the Institute of Mathematics of Jussieu 5(1), 81–99 (2006)

    Article  MathSciNet  Google Scholar 

  36. C. Maclachlan and A.W. Reid. The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, Vol. 219. Springer-Verlag, New York (2003).

  37. Martinez-Maure Y.: De nouvelles inégalités géométriques pour les hérissons. Archiv der Mathematik (Basel) 72(6), 444–453 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Maskit B.: Matrices for Fenchel-Nielsen coordinates. Ann. Acad. Sci. Fenn. Math. 26(2), 267–304 (2001)

    MathSciNet  MATH  Google Scholar 

  39. Mess G.: Lorentz spacetimes of constant curvature. Geometriae Dedicata 126, 3–45 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Näätänen M., Penner R.C.: The convex hull construction for compact surfaces and the Dirichlet polygon. Bulletin of the London Mathematical Society 23(6), 568–574 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. L. Nicolaescu. Lectures on the Geometry of Manifolds, 2nd edn. World Scientific, Hackensack (2007).

  42. Oliker V.I., Simon U.: Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature. J. Reine Angew. Math. 342, 35–65 (1983)

    MathSciNet  MATH  Google Scholar 

  43. Penner R.C.: The decorated Teichmüller space of punctured surfaces. Communications in Mathematical Physics 113(2), 299–339 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Ratcliffe. Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, Vol. 149, 2nd edn. Springer, New York (2006).

  45. R.T. Rockafellar and R. Wets. Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer-Verlag, Berlin (1998).

  46. T. Rockafellar. Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ. Reprint of the 1970 original, Princeton Paperbacks (1997).

  47. Rudin W.: Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York (1991)

    Google Scholar 

  48. Schlenker J.-M.: Small deformations of polygons and polyhedra. Transactions of the American Mathematical Society 359(5), 2155–2189 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Schneider R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  50. N. Trudinger and X.-J. Wang. The Monge-Ampère equation and its geometric applications. In: Handbook of Geometric Analysis. No. 1. Adv. Lect. Math. (ALM), Vol. 7. International Press, Somerville (2008), pp. 467–524.

  51. R.S. Varga. Matrix Iterative Analysis. Springer Series in Computational Mathematics, Vol. 27. Springer-Verlag, Berlin, expanded edition (2000).

Download references

Author information

Authors and Affiliations

  1. Departement of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088, 95000, Cergy-Pontoise, France

    François Fillastre

Authors
  1. François Fillastre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Fillastre.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fillastre, F. Fuchsian convex bodies: basics of Brunn–Minkowski theory. Geom. Funct. Anal. 23, 295–333 (2013). https://doi.org/10.1007/s00039-012-0205-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-012-0205-4

Keywords

Search

Navigation