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Transactions of the American Mathematical Society

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Boundary case of equality in optimal Loewner-type inequalities

Authors: Victor Bangert, Christopher Croke, Sergei V. Ivanov and Mikhail G. Katz
Journal: Trans. Amer. Math. Soc. 359 (2007), 1-17
MSC (2000): Primary 53C23; Secondary 57N65, 52C07
Published electronically: August 24, 2006
MathSciNet review: 2247879
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Abstract: We prove certain optimal systolic inequalities for a closed Riemannian manifold $ (X,\mathcal{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  $ \mathbb{T}^b$, which are area-decreasing (on $ b$-dimensional areas), with respect to suitable norms. These norms are the stable norm of $ \mathcal{G}$, the conformally invariant norm, as well as other $ L^p$-norms. Here we exploit $ L^p$-minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our 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. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under 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.

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Additional Information

Victor Bangert
Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany

Christopher Croke
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395

Sergei V. Ivanov
Affiliation: Steklov Mathematics Institute, Fontanka 27, RU-191011 St. Petersburg, Russia

Mikhail G. Katz
Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan 52900, Israel

Keywords: Abel-Jacobi map, conformal systole, deformation theorem, generalized degree, extremal lattice, free abelian cover, isoperimetric inequality, John ellipsoid, $L^p$-minimizing differential forms, Loewner inequality, perfect lattice, Riemannian submersion, stable systole, systolic inequality
Received by editor(s): June 8, 2004
Published electronically: August 24, 2006
Additional Notes: The first author was partially supported by DFG-Forschergruppe “Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis”
The second author was supported by NSF grant DMS 02-02536 and the Max-Planck-Institut für Mathematik Bonn
The third author was supported by grants CRDF RM1-2381-ST-02, RFBR 02-01-00090, and NS-1914.2003.1
The fourth author was supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03)
Article copyright: © Copyright 2006 American Mathematical Society

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