Prescribing curvature on compact surfaces with conical singularities

Troyanov, Marc

doi:10.1090/S0002-9947-1991-1005085-9

Prescribing curvature on compact surfaces with conical singularities

Author: Marc Troyanov
Journal: Trans. Amer. Math. Soc. 324 (1991), 793-821
MSC: Primary 53C45; Secondary 30F10, 58G30
DOI: https://doi.org/10.1090/S0002-9947-1991-1005085-9
MathSciNet review: 1005085
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Abstract: We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.

Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859
Melvyn S. Berger, Riemannian structures of prescribed Gaussian curvature for compact $2$-manifolds, J. Differential Geometry 5 (1971), 325–332. MR 295261
Jean-Pierre Bourguignon and Jean-Pierre Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987), no. 2, 723–736. MR 882712, DOI https://doi.org/10.1090/S0002-9947-1987-0882712-7
Pascal Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math. (2) 103 (1979), no. 4, 353–374 (French, with English summary). MR 548913
Pascal Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206 (French, with English summary). MR 749522, DOI https://doi.org/10.1016/0022-1236%2884%2990094-6
Sun-Yung A. Chang and Paul C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom. 27 (1988), no. 2, 259–296. MR 925123

R. Dautray and J. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Chapitre II, §3.2, Masson, Paris, 1987, pp. 335-349.

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443
Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 57, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787227
Jerry L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 343205, DOI https://doi.org/10.2307/1971012
Jerry L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317–331. MR 375153, DOI https://doi.org/10.2307/1970993
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI https://doi.org/10.1512/iumj.1971.20.20101
J. Moser, On a nonlinear problem in differential geometry, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 273–280. MR 0339258
Robert C. McOwen, Conformal metrics in ${\bf R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), no. 1, 97–104. MR 773395, DOI https://doi.org/10.1512/iumj.1985.34.34005
Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, DOI https://doi.org/10.1090/S0002-9939-1988-0938672-X

E. Picard, De l’intégration de l’équation $\Delta u = {e^u}$ sur une surface de Riemann fermée, Crelle’s J. 130 (1905), 243-258.

Marc Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), no. 1-2, 79–94 (French). MR 850552

---, Les surfaces riemanniennes à singularités coniques, Thèse, Université de Genève, 1987.

Marc Troyanov, Metrics of constant curvature on a sphere with two conical singularities, Differential geometry (Peñíscola, 1988) Lecture Notes in Math., vol. 1410, Springer, Berlin, 1989, pp. 296–306. MR 1034288, DOI https://doi.org/10.1007/BFb0086431