Il s'agit tout d'abord de déplier le polyèdre dans le plan, ce qui nous donne des coordonnées complexes correspondant aux sommets du polyèdre. L'aire du polyèdre est alors une forme quadratique de signature (1, n ? 3) en ces coordonnées, et comme on a fixé l'aire = 1, ceci fait naturellement apparaître le modèle de l'hyperboloïde du plan hyperbolique complexe ,
est-à-dire de polyèdres convexes dégénérés (contenus dans un plan), correspond à des sous-ensembles hyperboliques réels de l'espace de module de Thurston (J. Granier a étudié de tels sous-ensembles dans sa thèse ,
Thurston, ont décrit l'espace de module des polygones (convexes) du plan à n angles fixés : c'est un polyèdre de l'espace hyperbolique réel de dimension n ? 3. Là encore, c'est la forme d'aire qui fait apparaître la structure hyperbolique ,
L'idée générale de ce manuscrit étant d'étudier des métriques dont les singularités s'accumulent, on peut se poser la Harmonic function theory, 1992. ,
C ? -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom, vol.35, issue.2, pp.265-281, 1992. ,
Complex analysis An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, 1978. ,
Conformal invariants Topics in geometric function theory, Reprint of the 1973 original, 2010. ,
Convex polyhedra Translated from the, Zalgaller and appendices by L. A. Shor and Yu. A. Volkov, 1950. ,
Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math, vol.102, issue.2, pp.429-445, 1990. ,
General theory of irregular curves, volume 29 of Mathematics and its Applications (Soviet Series), 1989. ,
Intrinsic geometry of surfaces. Translated from the Russian by, J. M. Danskin. Translations of Mathematical Monographs, vol.15, 1967. ,
A course in metric geometry, Graduate Studies in Mathematics, vol.33, 2001. ,
DOI : 10.1090/gsm/033
Entropies et rigidit??s des espaces localement sym??triques de courbure strictement n??gative, Geometric and Functional Analysis, vol.114, issue.2, pp.731-799, 1995. ,
DOI : 10.1007/BF02684590
Les variétés Riemanniennes (1/4)-pincées, Ann. Scuola Norm. Sup. Pisa, vol.14, issue.3, pp.161-170, 1960. ,
Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, Journal of Differential Geometry, vol.5, issue.3-4, pp.325-332, 1971. ,
DOI : 10.4310/jdg/1214429996
A panoramic view of Riemannian geometry, 2003. ,
DOI : 10.1007/978-3-642-18245-7
Polygones du plan et polyedres hyperboliques, Geometriae Dedicata, vol.43, issue.2, pp.207-224, 1992. ,
DOI : 10.1007/BF00147868
Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not, issue.22, pp.1331-1369, 2005. ,
Topology and geometry, volume 139 of Graduate Texts in Mathematics, 1997. ,
Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products, The Annals of Mathematics, vol.144, issue.1, pp.189-237, 1996. ,
DOI : 10.2307/2118589
On the structure of spaces with Ricci curvature bounded below. I, Journal of Differential Geometry, vol.46, issue.3, pp.406-480, 1997. ,
DOI : 10.4310/jdg/1214459974
Scalar curvatures on S 2, Trans. Amer. Math. Soc, vol.303, issue.1, pp.365-382, 1987. ,
The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differential Equations, vol.1, issue.2, pp.205-229, 1993. ,
Finiteness Theorems for Riemannian Manifolds, American Journal of Mathematics, vol.92, issue.1, pp.61-74, 1970. ,
DOI : 10.2307/2373498
Prescribing Gaussian curvature on S2, Acta Mathematica, vol.159, issue.0, pp.215-259, 1987. ,
DOI : 10.1007/BF02392560
URL : http://doi.org/10.1007/bf02392560
Conformal deformation of metrics on S 2, J. Differential Geom, vol.27, issue.2, pp.259-296, 1988. ,
A compactness theorem for surfaces with bounded integral curvature, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01320662
Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup, vol.14, issue.43, pp.249-260, 1981. ,
A generalization of Berger's theorem on almost $\frac 14$-pinched manifolds. II, Journal of Differential Geometry, vol.26, issue.1, pp.101-13923, 1987. ,
DOI : 10.4310/jdg/1214441178
The heat equation shrinking convex plane curves, Journal of Differential Geometry, vol.23, issue.1 ,
DOI : 10.4310/jdg/1214439902
Riemannian geometry . Universitext, 2004. ,
URL : https://hal.archives-ouvertes.fr/hal-00002870
Some concepts and methods in Riemannian geometry, Differential geometry : Riemannian geometry, pp.1-22, 1990. ,
DOI : 10.1090/pspum/054.3/1216606
Curvature, diameter and betti numbers, Commentarii Mathematici Helvetici, vol.56, issue.1 ,
DOI : 10.1007/BF02566208
Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts]. CEDIC, vol.1, 1981. ,
Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications, 1996. ,
Lipschitz convergence of Riemannian manifolds, Pacific Journal of Mathematics, vol.131, issue.1, pp.119-141, 1988. ,
DOI : 10.2140/pjm.1988.131.119
URL : http://msp.org/pjm/1988/131-1/pjm-v131-n1-p07-s.pdf
The blow up analysis of the general curve shortening flow, Acta Math. Sin. (Engl. Ser.), issue.11, pp.272107-2128, 2011. ,
Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds, Rend. Mat. Appl, vol.17, issue.74, pp.569-605, 1997. ,
An introduction to Teichmüller spaces, 1992. ,
Geometrische Methoden zur Gewinnung von A-Priori-Schranken f???r harmonische Abbildungen, Manuscripta Mathematica, vol.19, issue.1, pp.27-77, 1982. ,
DOI : 10.1515/crll.1981.324.141
A convergence theorem for Riemannian manifolds and some applications, Nagoya Mathematical Journal, vol.49, pp.21-51, 1989. ,
DOI : 10.1007/BF01378068
URL : https://www.cambridge.org/core/services/aop-cambridge-core/content/view/23B5497F7DBE3C4F6E01AD9BA7B939F4/S0027763000001380a.pdf/div-class-title-a-convergence-theorem-for-riemannian-manifolds-and-some-applications-div.pdf
??ber Riemannsche Mannigfaltigkeiten mit positiver Kr??mmung, Commentarii Mathematici Helvetici, vol.35, issue.1, pp.47-54, 1961. ,
DOI : 10.1007/BF02567004
Curvature Functions for Compact 2-Manifolds, The Annals of Mathematics, vol.99, issue.1, pp.14-47, 1974. ,
DOI : 10.2307/1971012
Isometries of infinite dimensional Hilbert geometries, Journal of Topology and Analysis, vol.1, 2014. ,
DOI : 10.1353/ajm.2014.0036
An exotic deformation of the hyperbolic space, American Journal of Mathematics, vol.136, issue.5 ,
DOI : 10.1353/ajm.2014.0036
Ricci flow on surfaces with conic singularities, Analysis & PDE, vol.1, issue.4, pp.839-882, 2015. ,
DOI : 10.1007/s12220-010-9136-1
URL : http://arxiv.org/pdf/1306.6688
Convergence of Riemannian manifolds, Compositio Math, vol.62, issue.1, pp.3-16, 1987. ,
Convergence theorems in Riemannian geometry, Comparison geometry, pp.1993-94, 1997. ,
Riemannian geometry, volume 171 of Graduate Texts in Mathematics, 2006. ,
Extrinsic geometry of convex surfaces Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol.35, 1973. ,
Classification of almost quarter-pinched manifolds, Proc. Amer, pp.2437-2440, 2009. ,
DOI : 10.1090/S0002-9939-09-09802-5
URL : http://www.ams.org/proc/2009-137-07/S0002-9939-09-09802-5/S0002-9939-09-09802-5.pdf
Two-Dimensional Manifolds of Bounded Curvature, Encyclopaedia Math. Sci, vol.70, pp.3-163, 1993. ,
DOI : 10.1007/978-3-662-02897-1_1
On the conformal representation of Alexandrov surfaces In Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat, vol.83, pp.287-304, 2001. ,
Smoothening cone points with ricci flow, 2011. ,
Canonical smoothing of compact alexandrov surfaces via ricci flow. https ://arxiv.org/abs, 1204. ,
URL : https://hal.archives-ouvertes.fr/hal-00690909
Convex bodies : the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications, 1993. ,
Notes on shapes of polyhedra, 2013. ,
A sphere theorem for manifolds of positive Ricci curvature, Transactions of the American Mathematical Society, vol.275, issue.2 ,
DOI : 10.1090/S0002-9947-1983-0682734-1
The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Transactions of the American Mathematical Society, vol.351, issue.05, pp.1765-1801, 1999. ,
DOI : 10.1090/S0002-9947-99-02103-0
Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, pp.511-549 ,
Les surfaces euclidiennes à singularités coniques, Enseign. Math, vol.32, issue.212, pp.79-94, 1986. ,
Un principe de concentration-compacit?? pour les suites de surfaces Riemanniennes, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.8, issue.5, pp.419-441, 1991. ,
DOI : 10.1016/S0294-1449(16)30255-4
Prescribing curvature on compact surfaces with conical singularities, Transactions of the American Mathematical Society, vol.324, issue.2, pp.793-821, 1991. ,
DOI : 10.1090/S0002-9947-1991-1005085-9
URL : http://www.ams.org/tran/1991-324-02/S0002-9947-1991-1005085-9/S0002-9947-1991-1005085-9.pdf
On the moduli space of singular euclidean surfaces, Handbook of Teichmüller theory, pp.507-540, 2007. ,
DOI : 10.4171/029-1/13
Les surfaces à courbure intégrale bornée au sens d'alexandrov ,
Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds, Proc. Amer, pp.660-666, 1988. ,
DOI : 10.1090/S0002-9939-1988-0928999-X