Remarks on prescribing Gauss curvature

Xu, Xingwang; Yang, Paul C.

doi:10.1090/S0002-9947-1993-1087058-5

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.

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