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
Full-text PDF
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.
- [1] 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
- [2] Melvyn S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geometry 5 (1971), 325–332. MR 0295261
- [3] 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, https://doi.org/10.1090/S0002-9947-1987-0882712-7
- [4] 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
- [5] 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, https://doi.org/10.1016/0022-1236(84)90094-6
- [6] Sun-Yung A. Chang and Paul C. Yang, Conformal deformation of metrics on 𝑆², J. Differential Geom. 27 (1988), no. 2, 259–296. MR 925123
- [7] 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.
- [8] 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
- [9] 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
- [10] Jerry L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 0343205, https://doi.org/10.2307/1971012
- [11] 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 0375153, https://doi.org/10.2307/1970993
- [12] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 0301504, https://doi.org/10.1512/iumj.1971.20.20101
- [13] 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
- [14] Robert C. McOwen, Conformal metrics in 𝑅² with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), no. 1, 97–104. MR 773395, https://doi.org/10.1512/iumj.1985.34.34005
- [15] Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, https://doi.org/10.1090/S0002-9939-1988-0938672-X
- [16]
E. Picard, De l'intégration de l'équation
sur une surface de Riemann fermée, Crelle's J. 130 (1905), 243-258.
- [17] Marc Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), no. 1-2, 79–94 (French). MR 850552
- [18] -, Les surfaces riemanniennes à singularités coniques, Thèse, Université de Genève, 1987.
- [19] 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, https://doi.org/10.1007/BFb0086431
Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C45, 30F10, 58G30
Retrieve articles in all journals with MSC: 53C45, 30F10, 58G30
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1991-1005085-9
Keywords:
Surfaces,
curvature,
conical singularities
Article copyright:
© Copyright 1991
American Mathematical Society