A One-Dimensional Singular Non-CSC Extremal Kähler Metric can be Isometrically Imbedded into $${\mathbb {R}}^3$$ as a Weingarten Surface

Abstract

In this paper, we get a one-parameter family of local isometric immersions from a compact Riemann surface with a singular non-CSC extremal Kahler metric to $${\mathbb {R}}^3$$ , each of whom is a Weingarten surface. In fact, we can get explicit expressions of the mean curvatures in the family by the Gauss curvature of the metric.