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.

We consider the Monge-Ampère-type equation det(A+λg)= 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.

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},
     pages = {241--270},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {2},
     year = {2008},
     mrnumber = {2437027},
     zbl = {1192.53045},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2008_5_7_2_241_0/}
}
TY  - JOUR
AU  - Gursky, Matthew J.
TI  - A Monge-Ampère equation in conformal geometry
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2008
SP  - 241
EP  - 270
VL  - 7
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2008_5_7_2_241_0/
LA  - en
ID  - ASNSP_2008_5_7_2_241_0
ER  - 
%0 Journal Article
%A Gursky, Matthew J.
%T A Monge-Ampère equation in conformal geometry
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2008
%P 241-270
%V 7
%N 2
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2008_5_7_2_241_0/
%G en
%F 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://archive.numdam.org/item/ASNSP_2008_5_7_2_241_0/

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

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

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

[4] M. J. Gursky and J. A. Viaclovsky, A new variational characterization of three-dimensional space forms, Invent. Math. 145 (2001) 251-278. | MR | Zbl

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

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

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

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

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

[10] R. M. Schoen, 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 | Zbl

[11] W.-M. Sheng, N. S. Trudinger and X.-J. Wang, The Yamabe problem for higher order curvatures, J. Differential Geom. 77 (2007), 515-553. | MR | Zbl

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

[13] J. A. Viaclovsky, 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 | Zbl