Discrete & Computational Geometry

, Volume 40, Issue 4, pp 561–585

A Variational Proof of Alexandrov’s Convex Cap Theorem



We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures.


Convex cap Discrete Hilbert–Einstein functional Euclidean cone metric 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005) MATHGoogle Scholar
  2. 2.
    Bobenko, A., Izmestiev, I.: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447–505 (2008) MATHMathSciNetGoogle Scholar
  3. 3.
    Bobenko, A.I., Springborn, B.A.: Variational principles for circle patterns and Koebe’s theorem. Trans. Am. Math. Soc. 356(2), 659–689 (2004) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften, vol. 285. Springer, Berlin (1988) MATHGoogle Scholar
  5. 5.
    Cauchy, A.-L.: Sur les polygones et polyèdres, second mémoire. J. Ec. Polytech. 19, 87–98 (1813) Google Scholar
  6. 6.
    Darboux, G.: Leçons sur la théorie générale des surfaces. III, IV. Les Grands Classiques Gauthier-Villars. Éditions Jacques Gabay, Sceaux (1993) Google Scholar
  7. 7.
    Dehn, M.: Über die Starrheit konvexer Polyeder. Math. Ann. 77, 466–473 (1916) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fillastre, F.: Polyhedral realization of hyperbolic metrics with conical singularities on compact surfaces. Ann. Inst. Fourier (Grenoble) 57(1), 163–195 (2007) MATHMathSciNetGoogle Scholar
  9. 9.
    Fillastre, F., Izmestiev, I.: Hyperbolic cusps with convex polyhedral boundary. arXiv:0708.2666
  10. 10.
    Fillastre, F., Izmestiev, I.: Dual metrics of hyperbolic cusps with convex polyhedral boundary (in preparation) Google Scholar
  11. 11.
    Hodgson, C.D., Rivin, I.: A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111(1), 77–111 (1993) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Luo, F.: Rigidity of polyhedral surfaces. arXiv:math.GT/0612714
  13. 13.
    Milnor, J.: The Schläfli differential equality. In: Collected Papers, vol. 1, pp. x+295. Publish or Perish Inc., Houston (1994) Google Scholar
  14. 14.
    Sauer, R.: Projektive Sätze in der Statik des starren Körpers. Math. Ann. 110(1), 464–472 (1935) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Schlenker, J.-M.: On weakly convex star-shaped polyhedra. arXiv:0704.2901
  16. 16.
    Volkov, Yu.A.: Existence of convex polyhedra with prescribed development I. Vestn. Leningr. Univ. 15(19), 75–86 (1960) Google Scholar
  17. 17.
    Volkov, Yu.A.: An estimate for the deformation of a convex surface in dependence on the variation of its intrinsic metric. Ukr. Geom. Sb. 5–6, 44–69 (1968) Google Scholar
  18. 18.
    Volkov, Yu.A., Podgornova, E.G.: Existence of a convex polyhedron with prescribed development. Tašk. Gos. Pedagog. Inst. Uč. Zap. 85(3–54), 83 (1971) (Russian) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations