On isometric minimal immersion of a singular non-CSC extremal K$\ddot{a}$hler metric into 3-dimensional space forms

TL;DR

This paper proves that certain singular extremal Kähler metrics on compact Riemann surfaces cannot be minimally immersed into 3D space forms, highlighting limitations of such geometric embeddings.

Contribution

It demonstrates that non-CSC HCMU extremal Kähler metrics cannot be locally isometrically minimally immersed into 3D space forms, extending understanding of their geometric properties.

Findings

Non-CSC HCMU metrics cannot be minimally immersed into 3D space forms.

Such metrics cannot be immersed with constant mean curvature.

The results apply to all compact Riemann surfaces with these metrics.

Abstract

On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, by moving frames, we show that any non-CSC HCMU metric can not be isometrically minimal immersed into 3-dimensional real space forms even locally. In general, any non-CSC HCMU metric can not be isometrically immersed into 3-dimensional real space forms with constant mean curvature (CMC).