A Monge-Ampère equation in conformal geometry

Gursky, Matthew J.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, pp. 241-270.

We consider the Monge-Ampère-type equation $det(A+\lambda g)=\mathrm{const}.$, where $A$ is the Schouten tensor of a conformally related metric and $\lambda >0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

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