The Schottky problem on pants

Abstract

In this note, we consider the classical problem of Schottky of characterizing the set of period matrices which arise from all possible conformal structures on a fixed topological surface. Restricting to a planar surface with Euler characteristic − 1 - 1 , we find that a real symmetric 3 3 -by- 3 3 matrix arises as a period matrix if and only if the matrix has vanishing row sums, and the diagonal entries are positive and satisfy all three possible strict triangle inequalities. The technique of proof involves extremal and harmonic lengths of curve classes.