Prescribing curvature on compact surfaces with conical singularities
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- by Marc Troyanov
- Trans. Amer. Math. Soc. 324 (1991), 793-821
- DOI: https://doi.org/10.1090/S0002-9947-1991-1005085-9
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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.References
- 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, DOI 10.1007/978-1-4612-5734-9
- 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 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 10.1016/0022-1236(84)90094-6
- 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, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. 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, DOI 10.1090/cbms/057
- Jerry L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 343205, DOI 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 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 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 $\textbf {R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), no. 1, 97–104. MR 773395, DOI 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 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 10.1007/BFb0086431
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- 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