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Variational properties of the discrete Hilbert-Einstein functional. (English) Zbl 1477.53060

Summary (arXiv): This is a survey on rigidity and geometrization results obtained with the help of the discrete Hilbert-Einstein functional, written for the proceedings of the “Discrete Curvature” colloquium in Luminy.

MSC:

53C20 Global Riemannian geometry, including pinching
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
49Q10 Optimization of shapes other than minimal surfaces
53C24 Rigidity results
52B10 Three-dimensional polytopes
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry

References:

[1] Alexander D. Alexandrov, “Existence of a convex polyhedron and of a convex surface with a given metric”, Mat. Sbornik, N. Ser.11(53) (1942), p. 15-65, (Russian. English summary) · Zbl 0061.37603
[2] Michael T. Anderson, “Scalar curvature and the existence of geometric structures on 3-manifolds. I”, J. Reine Angew. Math.553 (2002), p. 125-182 | · Zbl 1023.53020 · doi:10.1515/crll.2002.096
[3] Michael T. Anderson, “Scalar curvature and the existence of geometric structures on 3-manifolds. II”, J. Reine Angew. Math.563 (2003), p. 115-195 | · Zbl 1066.53061 · doi:10.1515/crll.2003.080
[4] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10, Springer-Verlag, Berlin, 1987 · Zbl 1147.53001
[5] Wilhelm Blaschke & Gustav Herglotz, “Über die Verwirklichung einer geschlossenen Fläche mit vorgeschriebenem Bogenelement im Euklidischen Raum”, Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss. Abt.No.2 (1937), p. 229-230 · Zbl 0018.23501
[6] Alexander I. Bobenko & Ivan Izmestiev, “Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes”, Ann. Inst. Fourier (Grenoble)58 (2008) no. 2, p. 447-505 | · Zbl 1154.52005
[7] Eugenio Calabi, On compact, Riemannian manifolds with constant curvature. I, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, 1961, p. 155-180 · Zbl 0129.14102
[8] Jeff Cheeger, Werner Müller & Robert Schrader, “On the curvature of piecewise flat spaces”, Comm. Math. Phys.92 (1984) no. 3, p. 405-454 · Zbl 0559.53028
[9] Daryl Cooper & Igor Rivin, “Combinatorial scalar curvature and rigidity of ball packings”, Math. Res. Lett.3 (1996) no. 1, p. 51-60 | · Zbl 0868.51023 · doi:10.4310/MRL.1996.v3.n1.a5
[10] François Fillastre & Ivan Izmestiev, “Hyperbolic cusps with convex polyhedral boundary”, Geom. Topol.13 (2009) no. 1, p. 457-492 | · Zbl 1179.57026 · doi:10.2140/gt.2009.13.457
[11] David Glickenstein, “Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds”, J. Differential Geom.87 (2011) no. 2, p. 201-237 · Zbl 1243.52014
[12] Ivan Izmestiev, “The Colin de Verdière number and graphs of polytopes”, Israel J. Math.178 (2010), p. 427-444 · Zbl 1298.05204 · doi:10.1007/s11856-010-0070-5
[13] Ivan Izmestiev, “Examples of infinitesimally flexible 3-dimensional hyperbolic cone-manifolds”, J. Math. Soc. Japan63 (2011) no. 2, p. 581-598 · Zbl 1221.57029
[14] Ivan Izmestiev, “Infinitesimal rigidity of convex polyhedra through the second derivative of the Hilbert-Einstein functional”, Canad. J. Math.66 (2014) no. 4, p. 783-825 · Zbl 1303.52012
[15] Ivan Izmestiev & Jean-Marc Schlenker, “Infinitesimal rigidity of polyhedra with vertices in convex position”, Pacific J. Math.248 (2010) no. 1, p. 171-190 | · Zbl 1200.52014 · doi:10.2140/pjm.2010.248.171
[16] Norihito Koiso, “Nondeformability of Einstein metrics”, Osaka J. Math.15 (1978) no. 2, p. 419-433 · Zbl 0392.53030
[17] Rafe Mazzeo & Grégoire Montcouquiol, “Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra”, J. Differential Geom.87 (2011) no. 3, p. 525-576 · Zbl 1234.53014
[18] Tullio Regge, “General relativity without coordinates”, Nuovo Cimento19 (1961), p. 558-571
[19] Stefan Sechelmann, “Alexandrov’s Polyhedron Editor, Java Web Start application”, http://page.math.tu-berlin.de/ sechel/webstart/AlexandrovPolyhedron.jnlp
[20] André Weil, “On discrete subgroups of Lie groups”, Ann. of Math. (2)72 (1960), p. 369-384 · Zbl 0131.26602
[21] Hartmut Weiss, “The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than \(2 \pi\)”, Geom. Topol.17 (2013) · Zbl 1262.53032
[22] Hidehiko Yamabe, “On a deformation of Riemannian structures on compact manifolds”, Osaka Math. J.12 (1960), p. 21-37 | Copyright Cellule MathDoc 2018 · Zbl 0096.37201
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