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.

中文翻译：

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具有短闭合测地线的黎曼曲面的调和 1 型和雅可比的能量分布

我们研究了一个紧凑的双曲黎曼曲面 $S$ 上的谐波 1-形式的能量分布，其中一个短的闭合测地线被夹住。如果测地线将表面分成两部分，则 $S$ 的雅可比环面会发展为分裂的环面。如果测地线是非分离的，则 $S$ 的雅可比环面退化。这项工作的目的是深入了解这个过程，并根据初始表面 $S$ 和最终表面的几何数据进行估计，例如其注入半径和形成同源性基础的测地线长度。作为不变量，我们引入了新的辛矩阵族，以弥补在非紧致情况下全维 Gram 周期矩阵的不足。