C1,1 regularity of degenerate complex Monge–Ampère equations and some applications

Chu, Jianchun

doi:10.2140/apde.2021.14.1671

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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

DOI: 10.2140/apde.2021.14.1671

Abstract

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.

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

Vol. 14, No. 6, 2021