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

TL;DR

This paper investigates how harmonic 1-forms' energy distribution on hyperbolic Riemann surfaces changes as a short geodesic is pinched, affecting the Jacobian's structure and degeneracy.

Contribution

It introduces new estimates relating geometric data of Riemann surfaces to the energy distribution of harmonic forms during degeneration.

Findings

Jacobian torus splits when the geodesic separates the surface.

Jacobian torus degenerates when the geodesic is nonseparating.

New symplectic matrices are introduced to analyze degenerations.

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.