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Gaussian curvature in the negative case

Authors:	Wenxiong Chen Congming Li

Institution:	Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804 ; Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80039

Abstract:	In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case. In 1974, Kazdan and Warner considered the equation $\begin{displaymath}- \bigtriangleup u + \alpha = R(x)e^u, x \in M, \end{displaymath}$ on any compact two dimensional manifold with $\alpha < 0$ . They showed that there exists a number $\alpha_o$ , such that the equation is solvable for every $0 > \alpha > \alpha_o$ and it is not solvable for $\alpha < \alpha_o$ . Then one may naturally ask: Is the equation solvable for $\alpha = \alpha_o$ ? In this paper, we answer the question affirmatively. We show that there exists at least one solution for $\alpha = \alpha_o$ .

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