We prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions.

中文翻译：

---

复杂Monge–Ampère方程的几何估计

我们证明了一系列复杂的Monge–Ampère几何方程的均匀梯度和直径估计。这样的估计可用于研究复杂Monge-Ampère方程奇异解的几何正则性。我们还证明了非负Kodaira尺寸的Kähler流形上扭曲Kähler-Einstein度量的崩溃族的一致直径估计。