Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic. (arXiv:1810.05259v1 [math.DG])

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ that has a small separating closed geodesic. The result is applied to the question how close the Jacobian torus of $S$ comes to a torus that splits. The aim is to answer this and related questions in terms of geometric data of $S$ such as its injectivity radius and the lengths of geodesics that form a homology basis. This is version 1 of an extended paper in which also non separating small geodesics are considered. 查看全文>>