On the Nonexistence of a Holomorphic Isometric Immersion from a One-dimensional Singular Non-CSC Extremal Kähler Metric to ℂPN

Usually we call a one-dimensional singular non-CSC extremal Kähler metric HCMU metric. In this article, we will prove that there does not exist a local holomorphic isometric immersion from an HCMU metric to ℂ P N for N ≥ 2. The main method is to use the metric structure equations associated with the Frenet frame along a holomorphic curve in ℂ P N .