Counting components of moduli space of HCMU spheres via weighted trees

Abstract

HCMU surfaces are compact Riemann surfaces equipped with the Calabi extremal Kähler metric and a finite number of singularities. By using both the classical football decomposition introduced by Chen-Chen-Wu and the description of the geometric structure of HCMU surfaces by Lu-Xu, we can use weighted plane trees to characterize HCMU spheres with a single integral conical angle. Moreover, we obtain an explicit counting formula for the components of the moduli space of such HCMU spheres by enumerating some class of weighted plane trees.

26 pages, 5 figures