A Monge-Ampère equation in conformal geometry

Gursky, Matthew J.

Gursky, Matthew J.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 2, p. 241-270

Abstract

We consider the Monge-Ampère-type equation $det (A + λ g) = const .$ , where $A$ is the Schouten tensor of a conformally related metric and $λ > 0$ 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.

MR 2437027 | Zbl 1192.53045

Classification: 53A30

@article{ASNSP_2008_5_7_2_241_0,
     author = {Gursky, Matthew J.},
     title = {A Monge-Amp\`ere equation in conformal geometry},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {2},
     year = {2008},
     pages = {241-270},
     zbl = {1192.53045},
     mrnumber = {2437027},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2008_5_7_2_241_0}
}

Gursky, Matthew J. A Monge-Ampère equation in conformal geometry. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 2, pp. 241-270. http://www.numdam.org/item/ASNSP_2008_5_7_2_241_0/

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