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 variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian 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. The Jacobians in this paper are represented by Gram period matrices. 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.
Highlights
In this paper we make extensively use of the fact that when S is compact, in each cohomology class of closed 1-forms on S there is a unique harmonic form and that the latter is an energy minimizer, i.e. the harmonic form is the unique element in its cohomology class that has minimal energy
We denote by H1(S, Z) the first homology group and by H 1(S, R) the vector space of all real harmonic 1-forms on S
In the first part of this section we look at the energy distribution of harmonic forms that have zero periods on S2 respectively, S1
Summary
We construct mappings used to embed Riemann surfaces with a small geodesic into the limit surfaces with cusps obtained by the Fenchel–Nielsen construction. These mappings shall be used to compare the energies of corresponding harmonic forms with each other
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