A Monge-Ampère equation in conformal geometry
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 2, p. 241-270

We consider the Monge-Ampère-type equation $det\left(A+\lambda g\right)=\mathrm{const}.$, 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.

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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