Boundary case of equality in optimal Loewner-type inequalities
  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. 
Victor Bangert, Christopher Croke, Sergei V. Ivanov, Mikhail G. Katz