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.

@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/

[1] Kato constants in Riemannian geometry, Math. Res. Lett. 7 (2000), 245-261. | MR 1764320 | Zbl 1039.53033

,[2] Local estimates for some fully nonlinear elliptic equations, Int. Math. Res. Not. 55 (2005), 3403-3425. | MR 2204639 | Zbl 1159.35343

,[3] Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), 333-363. | MR 649348 | Zbl 0469.35022

,[4] A new variational characterization of three-dimensional space forms, Invent. Math. 145 (2001) 251-278. | MR 1872547 | Zbl 1006.58008

and ,[5] A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003), 131-154. | MR 2015262 | Zbl 1070.53018

and ,[6] Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J. 52 (2003), 399-419. | MR 1976082 | Zbl 1036.53025

and ,[7] Geometric inequalities on locally conformally flat manifolds, Duke Math. J. 124 (2004), 177-212. | MR 2072215 | Zbl 1059.53034

and ,[8] Conformal deformations of the smallest eigenvalue of the Ricci tensor, Amer. J. Math. 129 (2007), 499-526. | MR 2306044 | Zbl 1143.53033

and ,[9] Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 75-108. | MR 688919 | Zbl 0578.35024

,[10] Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations, (Montecatini Terme, 1987), Lecture Notes in Math., Vol. 1365, Springer, Berlin, 1989, 120-154. | MR 994021 | Zbl 0702.49038

,[11] The Yamabe problem for higher order curvatures, J. Differential Geom. 77 (2007), 515-553. | MR 2362323 | Zbl 1133.53035

, and ,[12] Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), 283-316. | MR 1738176 | Zbl 0990.53035

,[13] Conformal geometry and differential equations, to appear in: “Inspired by S. S. Chern: A Memorial Volume in Honor of a Great Mathematician”, P. Griffiths (ed.), Nankai Tracts in Mathematics, Vol. II, World Scientific, 2006. | MR 1764770 | Zbl 1142.53030

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