Gaussian curvature in the negative case

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 \[ - \bigtriangleup u + \alpha = R(x)e^u, \;\; x \in M, \] on any compact two dimensional manifold $M$ 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 $\alpha = \alpha _o$.