. De-façon-très-succincte, 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

. Le-cas-de-polygones-convexes, 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

. Ghys, 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

W. Fillastre-[-fil11-],-qui-fait-le-lien-entre-les-approches-de, C. Thurston-et-de, É. Bavard, and . Ghys, 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.

T. Michael, J. Anderson, and . Cheeger, C ? -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom, vol.35, issue.2, pp.265-281, 1992.

L. V. Ahlfors, Complex analysis An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, 1978.

L. V. Ahlfors, Conformal invariants Topics in geometric function theory, Reprint of the 1973 original, 2010.

]. A. Ale05 and . Alexandrov, Convex polyhedra Translated from the, Zalgaller and appendices by L. A. Shor and Yu. A. Volkov, 1950.

T. Michael and . Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math, vol.102, issue.2, pp.429-445, 1990.

Y. [. Alexandrov and . Reshetnyak, General theory of irregular curves, volume 29 of Mathematics and its Applications (Soviet Series), 1989.

V. [. Aleksandrov and . Zalgaller, Intrinsic geometry of surfaces. Translated from the Russian by, J. M. Danskin. Translations of Mathematical Monographs, vol.15, 1967.

D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol.33, 2001.
DOI : 10.1090/gsm/033

G. Besson, G. Courtois, and S. Gallot, 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

M. Berger, Les variétés Riemanniennes (1/4)-pincées, Ann. Scuola Norm. Sup. Pisa, vol.14, issue.3, pp.161-170, 1960.

M. S. Berger, 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

M. Berger, A panoramic view of Riemannian geometry, 2003.
DOI : 10.1007/978-3-642-18245-7

C. Bavard and É. Ghys, Polygones du plan et polyedres hyperboliques, Geometriae Dedicata, vol.43, issue.2, pp.207-224, 1992.
DOI : 10.1007/BF00147868

M. Burger, A. Iozzi, and N. Monod, Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not, issue.22, pp.1331-1369, 2005.

G. E. Bredon, Topology and geometry, volume 139 of Graduate Texts in Mathematics, 1997.

J. Cheeger and T. H. Colding, 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

J. Cheeger and T. H. Colding, 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

W. Xiong, C. , and W. Y. Ding, Scalar curvatures on S 2, Trans. Amer. Math. Soc, vol.303, issue.1, pp.365-382, 1987.

A. Sun-yung, M. J. Chang, P. C. Gursky, and . Yang, The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differential Equations, vol.1, issue.2, pp.205-229, 1993.

J. Cheeger, Finiteness Theorems for Riemannian Manifolds, American Journal of Mathematics, vol.92, issue.1, pp.61-74, 1970.
DOI : 10.2307/2373498

S. Chang and P. C. Yang, 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

A. Sun-yung, P. C. Chang, and . Yang, Conformal deformation of metrics on S 2, J. Differential Geom, vol.27, issue.2, pp.259-296, 1988.

C. Debin, A compactness theorem for surfaces with bounded integral curvature, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01320662

M. Dennis, J. L. Deturck, and . Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup, vol.14, issue.43, pp.249-260, 1981.

O. Durumeric, 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

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, vol.23, issue.1
DOI : 10.4310/jdg/1214439902

S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry . Universitext, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00002870

R. E. Greene, Some concepts and methods in Riemannian geometry, Differential geometry : Riemannian geometry, pp.1-22, 1990.
DOI : 10.1090/pspum/054.3/1216606

[. Gromov, Curvature, diameter and betti numbers, Commentarii Mathematici Helvetici, vol.56, issue.1
DOI : 10.1007/BF02566208

M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts]. CEDIC, vol.1, 1981.

H. Groemer, Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications, 1996.

H. [. Greene and . Wu, 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

R. Li, H. , and J. G. Bao, The blow up analysis of the general curve shortening flow, Acta Math. Sin. (Engl. Ser.), issue.11, pp.272107-2128, 2011.

E. Hebey and M. Herzlich, Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds, Rend. Mat. Appl, vol.17, issue.74, pp.569-605, 1997.

M. [. Imayoshi and . Taniguchi, An introduction to Teichmüller spaces, 1992.

J. Jost and H. Karcher, 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. Kasue, 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

W. Klingenberg, ??ber Riemannsche Mannigfaltigkeiten mit positiver Kr??mmung, Commentarii Mathematici Helvetici, vol.35, issue.1, pp.47-54, 1961.
DOI : 10.1007/BF02567004

J. L. Kazdan and F. W. Warner, Curvature Functions for Compact 2-Manifolds, The Annals of Mathematics, vol.99, issue.1, pp.14-47, 1974.
DOI : 10.2307/1971012

B. Lemmens, M. Roelands, and M. Wortel, Isometries of infinite dimensional Hilbert geometries, Journal of Topology and Analysis, vol.1, 2014.
DOI : 10.1353/ajm.2014.0036

N. Monod and P. Py, An exotic deformation of the hyperbolic space, American Journal of Mathematics, vol.136, issue.5
DOI : 10.1353/ajm.2014.0036

R. Mazzeo, Y. A. Rubinstein, and N. Sesum, 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

S. Peters, Convergence of Riemannian manifolds, Compositio Math, vol.62, issue.1, pp.3-16, 1987.

P. Petersen, Convergence theorems in Riemannian geometry, Comparison geometry, pp.1993-94, 1997.

P. Petersen, Riemannian geometry, volume 171 of Graduate Texts in Mathematics, 2006.

A. V. Pogorelov and R. I. , Extrinsic geometry of convex surfaces Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol.35, 1973.

P. Petersen and T. Tao, 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

Y. G. Reshetnyak, Two-Dimensional Manifolds of Bounded Curvature, Encyclopaedia Math. Sci, vol.70, pp.3-163, 1993.
DOI : 10.1007/978-3-662-02897-1_1

Y. G. Reshetnyak, On the conformal representation of Alexandrov surfaces In Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat, vol.83, pp.287-304, 2001.

[. Guallar, Smoothening cone points with ricci flow, 2011.

T. Richard, Canonical smoothing of compact alexandrov surfaces via ricci flow. https ://arxiv.org/abs, 1204.
URL : https://hal.archives-ouvertes.fr/hal-00690909

R. Schneider, Convex bodies : the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications, 1993.

R. Schwartz, Notes on shapes of polyhedra, 2013.

K. Shiohama, 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

T. Shioya, 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

P. William and . Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, pp.511-549

M. Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math, vol.32, issue.212, pp.79-94, 1986.

M. Troyanov, 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

M. Troyanov, 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

M. Troyanov, On the moduli space of singular euclidean surfaces, Handbook of Teichmüller theory, pp.507-540, 2007.
DOI : 10.4171/029-1/13

M. Troyanov, Les surfaces à courbure intégrale bornée au sens d'alexandrov

T. Yamaguchi, 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