We show existence and uniqueness of solutions to the Monge-Ampère equation on compact almost complex manifolds with non-integrable almost complex structure.
In this paper, we prove a C 1,1 estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this C 1,1 estimate, we show existence of C 1,1 solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above C 1,1 estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the C 1,1 regularity of geodesics in the space of Sasakian metrics.
In this paper, we prove the existence of solutions to the Fu-Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.2010 Mathematics Subject Classification. Primary: 58J05; Secondary: 53C55, 35J60.
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