Complex geodesics and complex Monge-Ampère equations with boundary singularity

2020/02/02 by Huang, Xiaojun, Xieping Wang, Wang, Xieping
Mathematics · #32F17 #32F45 #32H12 #32U35 #32W20 #35J96 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2002.00400

openalex publication_date 2020/02/02 · openalex created_date 2020/02/07 · openalex updated_date 2026/07/28

Abstract

We study complex geodesics and complex Monge-Ampère equations on bounded strongly linearly convex domains in \mathbb Cⁿ. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in \mathbb Cⁿ.

Citations

• Totally geodesic discs in strongly convex domains
• The pluricomplex Poisson kernel for strongly pseudoconvex domains

Related

• Complex Monge-Ampère equations with oblique boundary value
• An existence theorem for non-pluripolar complex Monge-Ampère type equations on hyperconvex domains
• The boundary case for complex Monge-Ampère type equations
• The Anisotropic Convexity of Domains and the Boundary Estimate for Two Monge-Ampère Equations
• Sharp boundary regularity for some degenerate-singular Monge-Ampère Equations on k-convex domain