  We prove certain optimal systolic inequalities for a closed Riemannian
manifold (X,g), depending on a pair of parameters, n and b. Here n is the
dimension of X, while b is its first Betti number. The proof of the
inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus
T^b, which are area-decreasing (on b-dimensional areas), with respect to
suitable norms. These norms are the stable norm of g, the conformally invariant
norm, as well as other L^p-norms. The case of equality is characterized in
terms of the criticality of the lattice of deck transformations of T^b, while
the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting
inequalities are actually nonvacuous follows from an isoperimetric inequality
of Federer and Fleming, in the assumption of the nonvanishing of the homology
class of the lift of the typical fiber of the Abel-Jacobi map to the maximal
free abelian cover.
