An HCMU surface is a compact Riemann surface carrying an extremal Kahler metric with finitely many conical singularities. Building on the football decomposition, most such surfaces are encoded uniquely by a data set that pairs discrete topological information (a weighted bi-colored graph on the surface obeying a balance equation) with continuous geometric parameters. This representation yields a unified proof of the angle constraints, an existence theorem for HCMU surfaces of any genus with a single saddle conical point, and a determination of the dimension of the moduli space (the number of independent continuous parameters) with fixed genus and conical angles via geometric deformations.
HCMU surfaces are compact Riemann surfaces carrying an extremal Kahler metric (whose Gaussian curvature satisfies a fourth-order elliptic PDE) together with finitely many conical singularities, first…