Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A best constant and the Gaussian curvature


Author: Chong Wei Hong
Journal: Proc. Amer. Math. Soc. 97 (1986), 737-747
MSC: Primary 58G30; Secondary 35B45, 53C20, 58E99
MathSciNet review: 845999
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For axisymmetric $ f \in {C^\infty }({S^2})$ we find conditions to make $ f$ the scalar curvature of a metric pointwise conformal to the standard metric of $ {S^2}$. Closely related to these results, we prove that in the inequality (Moser [8])

$\displaystyle \int_{{S^2}} {{e^u} \leq C{e^{\left\Vert {\nabla u} \right\Vert _... ...16\pi \quad }}\forall u \in H_1^2({S^2})} {\text{ with }}\int_{{S^2}} {u = 0} ,$

, the best constant $ C = {\text{Vol(}}{{\text{S}}^2}{\text{)}}$.

References [Enhancements On Off] (What's this?)

  • [1] Shing Tung Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 3–71. MR 645729
  • [2] 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
  • [3] J. Kazdan, Gaussian and scalar curvature, an update, Seminar on Differential Geometry (S. T. Yau, ed.), Princeton Univ. Press, Princeton, N. J., 1982, pp. 185-191.
  • [4] 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
  • [5] 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
  • [6] Thierry Aubin, Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32 (1979), no. 2, 148–174 (French, with English summary). MR 534672, 10.1016/0022-1236(79)90052-1
  • [7] Jerry L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 0343205
  • [8] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 0301504
  • [9] Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, 10.1002/cpa.3160360405
  • [10] E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen. I, Gewöhnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1967.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58G30, 35B45, 53C20, 58E99

Retrieve articles in all journals with MSC: 58G30, 35B45, 53C20, 58E99


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0845999-7
Keywords: Gaussian curvature, semilinear elliptic equation
Article copyright: © Copyright 1986 American Mathematical Society