Prescribing curvature with negative total curvature on open Riemann surfaces

Tang, Jun Jie

doi:10.1090/S0002-9939-1992-1110554-3

Prescribing curvature with negative total curvature on open Riemann surfaces
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Proc. Amer. Math. Soc. 116 (1992), 1023-1030 Request permission

Abstract:

On open Riemann surfaces, we obtain a necessary and sufficient condition for conformal metrics to have prescribed curvature and the desired singularities leading to negative total curvature.

References

Patricio Aviles, Conformal complete metrics with prescribed nonnegative Gaussian curvature in $\textbf {R}^2$, Invent. Math. 83 (1986), no. 3, 519–544. MR 827365, DOI 10.1007/BF01394420
Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal Gaussian curvature equation on $\textbf {R}^2$, Duke Math. J. 62 (1991), no. 3, 721–737. MR 1104815, DOI 10.1215/S0012-7094-91-06231-9
D. Hulin and M. Troyanov, Prescribing curvature on open surfaces, Math. Ann. 293 (1992), no. 2, 277–315. MR 1166122, DOI 10.1007/BF01444716
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, DOI 10.1090/cbms/057
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
Robert C. McOwen, Prescribed curvature and singularities of conformal metrics on Riemann surfaces, J. Math. Anal. Appl. 177 (1993), no. 1, 287–298. MR 1224820, DOI 10.1006/jmaa.1993.1258
Robert C. McOwen, Conformal metrics in $\textbf {R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), no. 1, 97–104. MR 773395, DOI 10.1512/iumj.1985.34.34005
Robert C. McOwen, On the equation $\Delta u+Ke^{2u}=f$ and prescribed negative curvature in $\textbf {R}^{2}$, J. Math. Anal. Appl. 103 (1984), no. 2, 365–370. MR 762561, DOI 10.1016/0022-247X(84)90133-1
Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, DOI 10.1090/S0002-9939-1988-0938672-X
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
Marc Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. MR 1005085, DOI 10.1090/S0002-9947-1991-1005085-9