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 | References | Similar Articles | Additional Information
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
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Additional Information
Keywords:
Surfaces,
curvature,
conical singularities
Article copyright:
© Copyright 1991
American Mathematical Society