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 the 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 equation. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.

中文翻译：

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简并复 Monge-Ampère 方程的 C1,1 正则性及一些应用

我们证明一个 $C^{1,1}$估计在紧致的几乎 Hermitian 流形上的复杂 Monge-Ampère 方程的解。使用这个$C^{1,1}$ 估计，我们证明存在 $C^{1,1}$退化的 Monge-Ampère 方程、相应的 Dirichlet 问题和奇异 Monge-Ampère 方程的解。我们还研究了复复形格林函数的奇点。另外，上面的证明$C^{1,1}$估计对一种复杂的 Monge-Ampère 型方程有效。作为几何应用，我们证明$C^{1,1}$ Sasakian 度量空间中测地线的规律性。