设 M 是一个紧致的 Kähler 流形，N 是高余维子流形。几年前，丘成桐教授提出研究如下问题：在非紧流形 M-N 上寻找完备的典则 Kähler 度量。为了解决这个问题，我们提出了 Poincaré-Mok-Yau（PMY）型度量的概念，并认为 PMY 型度量是解决这个问题的关键。本项目考虑 N 是由 k 个点组成的情形，主要内容分为三步。第一，分析 PMY 型常数量曲率 Kähler 度量存在性的几何与拓扑障碍。第二，在复 2 维 Toric 流形上解 PMY 型度量的常数量曲率方程。第三，在一般 Kähler 流形上解答丘成桐教授提出的问题。

Let M be a compact Kähler manifold and N its higher codimensional complex submanfold. Several years ago, Professor Shing-Tung Yau proposed a problem of how to find a complete canonical Kähler metric on the noncompact manifold M-N. To answer this problem, we introduced the concept of Poincaré-Mok-Yau type metric. In this project, we study this problem when N consists of k points. The strategy are divided into three parts. Fisrt, we will set up the geometric and toplogical obstructions for the existence of PMY type constant scalar Kähler metrics. Second, we will solve this problem on the toric manifolds of complex dimension 2. At last, we will solve Yau's problem in general case.