C1,1 regularity of degenerate complex Monge-Amp\`ere equations and some applications
Abstract
In this paper, we prove a C1,1 estimate for solutions of complex Monge-Amp\`ere equations on compact almost Hermitian manifolds. Using this C1,1 estimate, we show existence of C1,1 solutions to the degenerate Monge-Amp\`ere equations, the corresponding Dirichlet problems and the singular Monge-Amp\`ere equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above C1,1 estimate is valid for a kind of complex Monge-Amp\`ere type equations. As a geometric application, we prove the C1,1 regularity of geodesics in the space of Sasakian metrics.
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