ON ONE-DIMENSIONAL AND SINGULAR CALABI'S EXTREMAL METRICS WHOSE GAUSS CURVATURES HAVE NONZERO UMBILICAL HESSIANS

We consider, on compact Riemann surfaces, singular extremal metrics whose Gauss curvatures have nonzero umbilical Hessians, which are usually called HCMU metrics. The singular sets of these HCMU metrics consist of conical and cusp singularities, both of which are finitely many. We show that these metrics exist with the prescribed singularities if and only if so do certain meromorphic 1-forms on the Riemann surfaces, which only have simple poles with real residues and whose real parts are exact outside their poles.