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We study the energy distribution of harmonic 1-forms on a compact hyperbolicRiemann surface $S$ where a short closed geodesic is pinched. If the geodesicseparates the surface into two parts, then the Jacobian torus of $S$ developsinto a torus that splits. If the geodesic is nonseparating then the Jacobiantorus of $S$ degenerates. The aim of this work is to get insight into thisprocess and give estimates in terms of geometric data of both the initialsurface $S$ and the final surface, such as its injectivity radius and thelengths of geodesics that form a homology basis. As an invariant we introducenew families of symplectic matrices that compensate for the lack of fulldimensional Gram-period matrices in the noncompact case.

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