Multi-valued holomorphic functions and non-CSC extremal Kähler metrics with singularities on compact Riemann surfaces

An extremal Kähler metric with finite singularities on a compact Riemann surface is often called an HCMU metric. It can be regarded as a generalization of CSC (constant scalar curvature) metric. In this paper we will prove that any HCMU metric is a pullback of an HCMU metric on S2 with only two singularities (we call it football) by a multi-valued holomorphic function. Then we study the monodromy group of the multi-valued function.