  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].
