Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

Izmestiev, Ivan

Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 783-825

Résumé

The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of “warped polyhedra” with a fixed metric on the boundary.The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.In the spherical space and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert–Einstein functional and the volume, as well as between both kinds of rigidity.We review some of the related work and discuss directions for future research.

Détail

DOI : 10.4153/CJM-2013-031-9

Mots-clés : 52B99, 53C24, convex polyhedron, rigidity, Hilbert–Einstein functional, Minkowski theorem

Bibliographie

[1] [1] Aleksandrov, A. D., Zur Theorie der gemischten Volumina von konvexen Körpern. II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen. Rec. Math. Moscou, N. Ser. 2(1937), 1205–1238. Google Scholar

[2] [2] Aleksandrov, A. D., Existence of a convex polyhedron and of a convex surface with a given metric. Mat. Sbornik, N. S. 11(53)(1942), 15–65. Google Scholar

[3] [3] Aleksandrov, A. D., Convex polyhedra. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Google Scholar

[4] [4] Aleksandrov, A. D., Selected works. Part I. Classics of Soviet Mathematics, 4, Gordon and Breach Publishers, Amsterdam, 1996. Google Scholar

[5] [5] Blaschke, W., Ein Beweis für die Unverbiegbarkeit geschlossener konvexer Flächen. Gött. Nachr. (1912), 607–610. Google Scholar

[6] [6] Bobenko, A. I. and Izmestiev, I., Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2008), no. 2, 447–505. Google Scholar | DOI

[7] [7] Böhm, J. and Im Hof, H.-C., Flächeninhalt verallgemeinerter hyperbolischer Dreiecke. Geom. Dedicata 42(1992), no. 2, 223–233. Google Scholar | DOI

[8] [8] Calabi, E., On compact, Riemannian manifolds with constant curvature. I. Proc. Sympos. Pure Math., III, American Mathematical Society, Providence, RI, 1961, pp. 155–180. Google Scholar

[9] [9] Casson, A., An example of weak non-rigidity for cone manifolds with vertices. Talk at the Third MSJ regional workshop, Tokyo, 1998. Google Scholar

[10] [10] Cauchy, A.-L., Sur les polygones et polyèdres, second mémoire. J. de l’Ecole Polytechnique 19(1813), 87–98. Google Scholar

[11] [11] Cheeger, J., A vanishing theorem for piecewise constant curvature spaces. In: Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math., 1201, Springer, Berlin, 1986, pp. 33–40. Google Scholar | DOI

[12] [12] Cheeger, J.,Müller, W., and Schrader, R., On the curvature of piecewise flat spaces. Comm. Math. Phys. 92(1984), no. 3, 405–454. Google Scholar | DOI

[13] [13] Cho, Y. and Kim, H., The analytic continuation of hyperbolic space. Geom. Dedicata 161(2012), 129–155. Google Scholar | DOI

[14] [14] Crapo, H. and Whiteley, W., Statics of frameworks and motions of panel structures, a projective geometric introduction. Structural Topology 1982, no. 6, 43–82. Google Scholar

[15] [15] Dehn, M., Über die Starrheit konvexer Polyeder. Math. Ann. 77(1916), no. 4, 466–473. Google Scholar | DOI

[16] [16] Dodziuk, J., Finite-difference approach to the Hodge theory of harmonic forms. Amer. J. Math. 98(1976), no. 1, 79–104. Google Scholar | DOI

[17] [17] Fillastre, F. and Izmestiev, I., Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13(2009), no. 1, 457–492. Google Scholar | DOI

[18] [18] Fillastre, F., Gauss images of hyperbolic cusps with convex polyhedral boundary. Trans. Amer. Math. Soc. 363(2011), no. 10, 5481–5536. Google Scholar | DOI

[19] [19] Filliman, P., Rigidity and the Alexandrov-Fenchel inequality. Monatsh. Math. 113(1992), no. 1, 1–22. Google Scholar | DOI

[20] [20] Forman, R., Bochner's method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(2003), no. 3, 323–374. Google Scholar | DOI

[21] [21] Glickenstein, D., Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds. J. Differential Geom. 87(2011), no. 2, 201–237. Google Scholar | DOI

[22] [22] Herglotz, G., Über die Steinersche Formel für Parallelflüchen. Abh. Math. Sem. Hansischen Univ. 15(1943), 165–177. Google Scholar | DOI

[23] [23] Hodgson, C. D. and Kerckhoff, S. P., Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom. 48(1998), no. 1, 1–59. Google Scholar | DOI

[24] [24] Izmestiev, I., A variational proof of Alexandrov's convex cap theorem. Discrete Comput. Geom. 40(2008), no. 4, 561–585. Google Scholar | DOI

[25] [25] Izmestiev, I., Projective background of the infinitesimal rigidity of frameworks. Geom. Dedicata 140(2009), 183–203. Google Scholar | DOI | DOI

[26] [26] Izmestiev, I., Examples of infinitesimally flexible 3-dimensional hyperbolic cone-manifolds. J. Math. Soc. Japan 63(2011), no. 2, 581–598. Google Scholar | DOI

[27] [27] Izmestiev, I., Infinitesimal rigidity of smooth convex surfaces through the second derivative of the Hilbert-Einstein functional. Dissertationes Math. 492(2013), 58 pp. Google Scholar | DOI

[28] [28] Izmestiev, I. and Schlenker, J.-M., Infinitesimal rigidity of polyhedra with vertices in convex position. Pacific J. Math. 248(2010), no. 1, 171–190. Google Scholar | DOI

[29] [29] Koiso, N., Nondeformability of Einstein metrics. Osaka J. Math. 15(1978), no. 2, 419–433. Google Scholar

[30] [30] Lee, C.W., P.L.-spheres, convex polytopes, and stress. Discrete Comput. Geom. 15(1996), no. 4, 389–421. Google Scholar | DOI

[31] [31] Legendre, A.-M., Eláments de gáomátrie. Firmin Didot, 1974. Google Scholar

[32] [32] Mazzeo, R. and Montcouquiol, G., Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra. J. Differential Geom. 87(2011), no. 3, 525–576. Google Scholar | DOI

[33] [33] McMullen, P., Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Cambridge Philos. Soc. 78(1975), no. 2, 247–261. Google Scholar | DOI

[34] [34] McMullen, P., Weights on polytopes. Discrete Comput. Geom. 15(1996), no. 4, 363–388. Google Scholar | DOI

[35] [35] Milnor, J., The Schläfli differential equality. In: Collected papers. Vol. 1, Publish or Perish Inc., Houston, TX, 1994, pp. 281–295. Google Scholar

[36] [36] Minkowski, H., Volumen und Oberfläche. Math. Ann. 57(1903), no. 4, 447–495. Google Scholar | DOI

[37] [37] Pak, I. M., A short proof of the rigidity of convex polytopes. Sibirsk. Mat. Zh. 47(2006), no. 4, 859–864; translation in Siberian Math. J. 47(2006), no. 4, 710–713. Google Scholar

[38] [38] Pogorelov, A. V., A new proof of rigidity of convex polyhedra. (Russian) Uspehi Mat. Nauk (N.S.) 11(1956), 207–208. Google Scholar

[39] [39] Pogorelov, A. V., Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs, 35, American Mathematical Society, Providence, RI, 1973. Google Scholar

[40] [40] Regge, T., General relativity without coordinates. Nuovo Cimento 19(1961), 558–571. Google Scholar | DOI | DOI

[41] [41] Rivin, I., Geometry of polyhedra in hyperbolic 3–space. Ph.D. thesis, Princeton University, 1986. Google Scholar

[42] [42] Roth, B. and Whiteley, W., Tensegrity frameworks. Trans. Amer. Math. Soc. 265(1981), no. 2, 419–446. Google Scholar | DOI

[43] [43] Santalá, L. A., Integral geometry and geometric probability. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. Google Scholar

[44] [44] Sauer, R., Darboux-Kranz verknickbarer Vierecksgitter. Arch. Math. 6(1955), 180–184. Google Scholar | DOI

[45] [45] Schlenker, J.-M., Hyperbolic manifolds with convex boundary. Invent. Math. 163(2006), no. 1, 109–169. Google Scholar | DOI

[46] [46] Schlenker, J.-M., Small deformations of polygons and polyhedra. Trans. Amer. Math. Soc. 359(2007), no. 5, 2155–2189. Google Scholar | DOI

[47] [47] Schlenker, J.-M., On weakly convex star-shaped polyhedra. Discrete Math. 309(2009), no. 20, 6139–6145. Google Scholar | DOI

[48] [48] Schneider, R., Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. Google Scholar

[49] [49] Steiner, J., Über parallele Flächen. Monatsbericht der Akademie der Wissenschaften zu Berlin, 1840, 114–118. Google Scholar

[50] [50] Suárez-Peiró, E., A Schläfli differential 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. Pacific J. Math. 194(2000), no. 1, 229–255. Google Scholar | DOI

[51] [51] Tay, T.-S., White, N., and Walter, W., Skeletal rigidity of simplicial complexes. I, II. European J. Combin. 16(1995), no. 4, 381–403; 502–523. Google Scholar | DOI

[52] [52] Trushkina, V. I., A theorem on the coloring and rigidity of a convex polyhedron. (Russian) Ukrain. Geom. Sb. 24(1981), 116–122. Google Scholar

[53] [53] Y. A. Volkov, Existence of a polyhedron with a given development. Ph.D. thesis, Leningrad State University, 1955. Google Scholar

[54] [54] Volkov, Y. A., On deformations of a convex polyhedral angle. (Russian) Uspehi Mat. Nauk (N.S.) 11(1956), no. 5(71), 209–210. Google Scholar

[55] [55] Volkov, Y. A., An estimate of the deformation of a convex surface as a function of the change in its intrinsic metric. (Russian) Ukrain. Geometr. Sb. Vyp. 5–6(1968), 44–69. Google Scholar

[56] [56] Volkov, Y. A. and Podgornova, E. G., Existence of a convex polyhedron with prescribed development. (Russian) Taškent. Gos. Ped. Inst. Učen. Zap. 85(1971), 3–54, 83. Google Scholar

[57] [57] Weil, A., On discrete subgroups of Lie groups. Ann. of Math. (2) 72(1960), 369–384. Google Scholar | DOI

[58] [58] Weiss, H., The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2π. Geom. Topol. 17(2013), no. 1, 329–367. Google Scholar | DOI

[59] [59] Weyl, H., Über die Starrheit der Eifla¨chen und konvexer Polyeder. Berl. Ber. 1917(1917), 250–266. Google Scholar

[60] [60] Whiteley, W., Rigidity and polarity. I. Statics of sheet structures. Geom. Dedicata 22(1987), no. 3, 329–362. Google Scholar