The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics

Details The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics

2012-02-03

arxiv.org/abs/1202.0782v1

Mathematics

Differential Geometry

25 pages, 8 figures

Scientific paper

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 we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. We give upper and lower bounds for all 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 (scg). We first present a theoretical approach that relies on the premise that these cut loci can be calculated. Then we give an example where our upper bound is sharp. Finally we give practical estimates based on the geometry of Q-pieces that contain a canonical homology basis. The methods developed here have been applied to surfaces that contain small non-separating scgs in [BMMS].

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