Conformal metrics with prescribed Gaussian curvature on 𝑆²

Cheng, Kuo-Shung; Smoller, Joel A.

doi:10.1090/S0002-9947-1993-1087053-6

Conformal metrics with prescribed Gaussian curvature on $S^ 2$
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by Kuo-Shung Cheng and Joel A. Smoller PDF

Trans. Amer. Math. Soc. 336 (1993), 219-251 Request permission

Abstract:

We consider on ${S^2}$ the problem of which functions $K$ can be the scalar curvature of a metric conformal to the standard metric on ${S^2}$. We assume that $K$ is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on $k$ (which are easy to check), in order that the problem be solvable.

References

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
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
Kuo-Shung Cheng and Jenn-Tsann Lin, On the elliptic equations $\Delta u=K(x)u^\sigma$ and $\Delta u=K(x)e^{2u}$, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639–668. MR 911088, DOI 10.1090/S0002-9947-1987-0911088-1
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 and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 343205, DOI 10.2307/1971012
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
Wei Ming Ni, On the elliptic equation $\Delta u+K(x)e^{2u}=0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), no. 2, 343–352. MR 656628, DOI 10.1007/BF01389399
D. H. Sattinger, Conformal metrics in $\textbf {R}^{2}$ with prescribed curvature, Indiana Univ. Math. J. 22 (1972/73), 1–4. MR 305307, DOI 10.1512/iumj.1972.22.22001