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Boundary case of equality in optimal Loewner-type inequalities. (English) Zbl 1122.53020

In the paper under review, the authors prove certain optimal systolic inequalities for a closed Riemannian manifold \((X,G)\), depending on a pair of parameters, namely, the dimension \(n\) of \(X\) and its first Betty number \(b\).
The authors obtain a conformally invariant analogue of an optimal inequality of this kind, as proved by M. Gromov under the assumption \(n=b\) (compare Theorems 1.1. and 1.2.). Moreover, they precisely examine the critical cases where equality is obtained, in both, Gromov’s systolic inequality and its conformal analogue (compare theorems 1.3. and 1.4.).
The main results, as stated in Theorems 3.6. and 3.8, further generalize the conformal analogue of Gromov’s systolic inequality to the case \(n\geq b\).
The proof of the inequalities involves constructing Abel-Jacobi maps from \(X\) to its Jacobi torus \(\mathbb{T}^b\), 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. Here the authors exploit \(L^p\)-minimizing differential \(1\)-forms in cohomology classes. They characterize the case of equality in their optimal inequalities, in terms of the criticality of the lattice of deck transformations of \(\mathbb{T}^b\), while the Abel-Jacobi map is a harmonic Riemannian submersion.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57N65 Algebraic topology of manifolds
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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