Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 693-721

THE JACOBIAN OF A RIEMANN SURFACE AND THE GEOMETRY OF THE CUT LOCUS OF SIMPLE CLOSED GEODESICS

Bjoern Muetzel

Dartmouth College, Department of Mathematics
Hanover, NH 03755, U.S.A.; bjorn.mutzel 'at' gmail.com

Abstract. 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].

2010 Mathematics Subject Classification: Primary 14H40, 14H42, 30F15, 30F45.

Key words: Riemann surfaces, Jacobians, harmonic forms, energy, hyperbolic geometry.

Reference to this article: B. Muetzel: The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics. Ann. Acad. Sci. Fenn. Math. 42 (2017), 693-721.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4242