The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics
| Autor: | Bjoern Muetzel |
|---|---|
| Přispěvatelé: | GTA, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Alexander von Humboldt Foundation |
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: |
Mathematics - Differential Geometry
Abelian variety period matrix Geodesic General Mathematics Geometry Cut locus Homology (mathematics) 01 natural sciences symbols.namesake 0103 physical sciences FOS: Mathematics 14H40 14H42 30F15 30F45 Compact Riemann surface 0101 mathematics Mathematics energy estimates Riemann surface 010102 general mathematics Complex torus Riemann surfaces Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Jacobian matrix and determinant symbols 010307 mathematical physics 14H40 14H42 30F15 30F45 Jacobian |
| Popis: | To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the entries of the period Gram matrix with respect to a suitable homology basis. These bounds depend on the geometry of the cut locus of non-separating simple closed geodesics. Assuming that the cut loci can be calculated, a theoretical approach is presented followed by an example where the upper bound is sharp. Finally we give practical estimates based on the Fenchel-Nielsen coordinates of surfaces of signature (1,1), or Q-pieces. The methods developed here have been applied to surfaces that contain small non-separating simple closed geodesics in [BMMS]. Comment: 32 pages, 10 figures, Annales Academi{\ae} Scientiarum Fennic{\ae} Mathematica |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|