Remarks on prescribing Gauss curvature
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- by Xingwang Xu and Paul C. Yang PDF
- Trans. Amer. Math. Soc. 336 (1993), 831-840 Request permission
Abstract:
We study the nonlinear partial differential equation for the problem of prescribing Gauss curvature $K$ on ${S^2}$ . We give an example of a rotationally symmetric $K$ for which the Kazdan-Warner obstruction is satisfied but the equation has no rotationally symmetric solution. On the other hand, we give a simple sufficient condition for solvability of the equation when $K$ is rotationally symmetric. Finally we give a sufficient condition for solvability when $K$ is not necessarily rotationally symmetric.References
- 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. Functional Analysis 32 (1979), no. 2, 148–174 (French, with English summary). MR 534672, DOI 10.1016/0022-1236(79)90052-1
- Abbas Bahri and Jean-Michel Coron, Une théorie des points critiques à l’infini pour l’équation de Yamabe et le problème de Kazdan-Warner, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 15, 513–516 (French, with English summary). MR 792378
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- Jean-Pierre Bourguignon, Invariants intégraux fonctionnels pour des équations aux dérivées partielles d’origine géométrique, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 100–108 (French). MR 863748, DOI 10.1007/BFb0076623
- Sun-Yung Alice Chang and Paul C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math. 159 (1987), no. 3-4, 215–259. MR 908146, DOI 10.1007/BF02392560
- 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
- Sun-Yung A. Chang and Paul C.-P. Yang, Isospectral conformal metrics on $3$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 1, 117–145. MR 1015647, DOI 10.1090/S0894-0347-1990-1015647-3 —, A perturbation result in prescribing scalar curvature on ${S^n}$ , preprint, 1990.
- Wen Xiong Chen and Wei Yue Ding, Scalar curvatures on $S^2$, Trans. Amer. Math. Soc. 303 (1987), no. 1, 365–382. MR 896027, DOI 10.1090/S0002-9947-1987-0896027-4 K.-S. Cheng and J. Smoller, On conformal metrics with prescribed Gaussian curvature on ${S^2}$ , preprint, 1990.
- José F. Escobar and Richard M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), no. 2, 243–254. MR 856845, DOI 10.1007/BF01389071 Z.-C. Han, Ph.D. thesis, New York University, 1989.
- Chong Wei Hong, A best constant and the Gaussian curvature, Proc. Amer. Math. Soc. 97 (1986), no. 4, 737–747. MR 845999, DOI 10.1090/S0002-9939-1986-0845999-7
- Jerry L. Kazdan, An isoperimetric inequality and Wiedersehen manifolds, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 143–157. MR 645734
- 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
- Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435
- 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
- E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys. 86 (1982), no. 3, 321–326. MR 677001
- B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5 R. Schoen and D. Zhang, private communication. D. Zhang, Ph.D. Thesis, Stanford University, 1990.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 831-840
- MSC: Primary 53C21; Secondary 35J60
- DOI: https://doi.org/10.1090/S0002-9947-1993-1087058-5
- MathSciNet review: 1087058