Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

Abstract

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $S$ degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface $S$ and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.