Vol. 14, No. 6, 2021

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$C^{1,1}$ regularity of degenerate complex Monge–Ampère equations and some applications

Jianchun Chu

Vol. 14 (2021), No. 6, 1671–1700
Abstract

We prove a C1,1 estimate for solutions of complex Monge–Ampère equations on compact almost Hermitian manifolds. Using this C1,1 estimate, we show the existence of C1,1 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 C1,1 estimate is valid for a kind of complex Monge–Ampère-type equation. As a geometric application, we prove the C1,1 regularity of geodesics in the space of Sasakian metrics.

Keywords
degenerate complex Monge–Ampère equation, $C^{1,1}$ estimate
Mathematical Subject Classification 2010
Primary: 32W20
Secondary: 35J70, 53C15, 32Q60, 35J75, 53C25
Milestones
Received: 12 December 2018
Revised: 30 March 2019
Accepted: 16 March 2020
Published: 7 September 2021
Authors
Jianchun Chu
Institute of Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing
China