Complex Monge-Ampere equations and totally real submanifolds

2009/10/09 by Bo Guan, Qun Li, Guan, Bo +1 · 5 citations
Mathematics · #32W20 #35J25 #53C55 #58J05 #58J32 #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0910.1851

openalex publication_date 2009/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kaehler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampere (\em HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kaehler metrics.

Citations

• Hermitian Curvature Flow

Cited by

• Uniform L^∞ estimates for complex hessian equations on compact Hermitian manifolds
• The Critical LYZ equation in Kähler Geometry
• The complex Monge-Ampere equation and an application to uniformisation of surfaces
• Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds
• Degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and applications II

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