We study complex geodesics and complex Monge–Ampère equations on bounded strongly linearly convex domains in \(\mathbb C^n\). 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 \({\mathbb {C}}^n\).

我们研究\（\ mathbb C ^ n \）中有界的强线性凸域上的复杂测地线和复杂的Monge-Ampère方程。更具体地说，当边界的规则性最小时，我们证明了复杂的测地线具有规定的边界值和方向的唯一性。1990年代初的第一作者证明了这种复杂的大地测量学的存在，但其独特性尚待解决。基于此处证明的存在性和唯一性以及其他先前获得的结果，我们求解了具有规定的边界奇点的齐次复杂Monge–Ampère方程，这是Bracci等人首先考虑的。在\（{\ mathbb {C}} ^ n \）中的光滑边界强凸域上。