Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic. (English) Zbl 1473.30023
Let \(S_t\) be a continuous family of compact hyperbolic Riemann surfaces of genus \(g\ge 2\). Here \(t\) is the length of the shortest closed geodesic of \(S_t\). The authors study the energy distribution of the harmonic 1-forms on \(S_t\) as \(t\to 0\). To describe the family \(S_t\) more exactly, two approaches are used; they are based on the Fenchel-Nielsen coordinates and the grafting construction.
Two fundamentally different cases are studied. If the shortest closed geodesic separates \(S_t\) into two parts, then we have the separating case; the opposite situation is called non-separating case. In the separating case, the limiting surface is a noncompact hyperbolic surface of genus \(g - 1\) with two cusps. In the separating case, the authors obtain a pair of noncompact hyperbolic surfaces, each with one cusp. The asymptotical behavior is described in terms of the Gram-period matrix corresponding to the canonical cohomology basis in the space of \(1\)-differential forms on \(S_t\).
Two fundamentally different cases are studied. If the shortest closed geodesic separates \(S_t\) into two parts, then we have the separating case; the opposite situation is called non-separating case. In the separating case, the limiting surface is a noncompact hyperbolic surface of genus \(g - 1\) with two cusps. In the separating case, the authors obtain a pair of noncompact hyperbolic surfaces, each with one cusp. The asymptotical behavior is described in terms of the Gram-period matrix corresponding to the canonical cohomology basis in the space of \(1\)-differential forms on \(S_t\).
Reviewer: Samyon R. Nasyrov (Kazan)
MSC:
| 30F10 | Compact Riemann surfaces and uniformization |
| 14H40 | Jacobians, Prym varieties |
| 14H42 | Theta functions and curves; Schottky problem |
| 30F15 | Harmonic functions on Riemann surfaces |
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
Keywords:
Riemann surface; Jacobian variety; harmonic 1-form; Teichmüller space; Gram-period matrix; Fenchel-Nielsen coordinates; grafting constructionReferences:
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