Abstract
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.
Funding statement: Research supported in part by National Science Foundation grants DMS-1406124 and DMS-1710500.
Acknowledgements
This is part of the thesis of Xin Fu at Rutgers University, and he would like to thank the Department of Mathematics for its generous support. Bin Guo would like to thank Professor D. H. Phong for many stimulating discussions and his constant support and encouragement. The authors thank Valentino Tosatti for helpful comments on the earlier draft. The authors also want to thank the referee for many helpful and valuable suggestions.
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