不变度量是多复变与复几何中的重要研究对象。Kobayashi度量作为经典Poincaré度量的高维推广，是多复变与复几何中最重要的不变度量之一。上世纪80年代，Lempert关于C^n中有界强线性凸域上Kobayashi度量与复测地线的重要工作建立了不变度量与复Monge-Ampère方程的深刻联系，在多复变、CR几何中具有广泛的应用。现在，该工作通常被称为Lempert复测地线理论。. 本项目旨在解决Lempert复测地线理论方面的遗留问题，进一步发掘复测地线与复Monge-Ampère方程的联系。我们将证明C^n中有界强线性凸域上具有预定边值和方向的复测地圆盘的唯一性，进而求解一个带边界奇性的齐次复Monge-Ampère方程。我们的结果将有望解决相关的公开问题。

Invariant metrics are important research objects in several complex variables and complex geometry. As a high-dimensional generalization of the classic Poincaré metric, the Kobayashi metric is one of the most important invariant metrics in several complex variables and complex geometry. In the 1980s, Lempert's important work on the Kobayashi metric and complex geodesics of strongly linearly convex domains in C^n established a deep connection between invariant metrics and complex Monge-Ampère equations, which has found extensive applications in several complex variables and CR geometry. This work is now usually referred to as Lempert complex geodesic theory.. This project aims to solve the remaining problems in Lempert complex geodesic theory, and further to explore the connection between complex geodesics and complex Monge-Ampère equations. We will prove the uniqueness of complex geodesic discs with prescribed boundary value and direction on bounded strongly linear convex domains in C^n, and solve a homogeneous complex Monge-Ampère equation with boundary singularity. Our results hopefully resolve some related open problems.