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

Abstract

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

37 pages, v2: added a geometric application ($C^{1,1}$ regularity of geodesics in the space of Sasakian metrics); added references; corrected typos

References in corpus (2)

• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• Quasiplurisubharmonic Green functions