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

In: Transactions of the American Mathematical Society, Vol. 359, No. 1, 01.01.2007, p. 1-17.

Research output: Contribution to journal › Article › peer-review

Harvard

Bangert, V, Croke, C, Ivanov, SV & Katz, MG 2007, 'Boundary case of equality in optimal loewner-type inequalities', Transactions of the American Mathematical Society, vol. 359, no. 1, pp. 1-17. https://doi.org/10.1090/S0002-9947-06-03836-0

APA

Bangert, V., Croke, C., Ivanov, S. V., & Katz, M. G. (2007). Boundary case of equality in optimal loewner-type inequalities. Transactions of the American Mathematical Society, 359(1), 1-17. https://doi.org/10.1090/S0002-9947-06-03836-0

Vancouver

Bangert V, Croke C, Ivanov SV, Katz MG. Boundary case of equality in optimal loewner-type inequalities. Transactions of the American Mathematical Society. 2007 Jan 1;359(1):1-17. https://doi.org/10.1090/S0002-9947-06-03836-0

Author

Bangert, Victor ; Croke, Christopher ; Ivanov, Sergei V. ; Katz, Mikhail G. / Boundary case of equality in optimal loewner-type inequalities. In: Transactions of the American Mathematical Society. 2007 ; Vol. 359, No. 1. pp. 1-17.

BibTeX

@article{8b204ca16407408a9fc8024a7e16b5b3,
title = "Boundary case of equality in optimal loewner-type inequalities",
abstract = "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 Tb, 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 Lp-norms. Here we exploit Lp-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 Tb, 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.",
keywords = "Abel-Jacobi map, Conformal systole, Deformation theorem, Extremal lattice, Free abelian cover, Generalized degree, Isoperimetric inequality, John ellipsoid, Loewner inequality, Lp-minimizing differential forms, Perfect lattice, Riemannian submersion",
author = "Victor Bangert and Christopher Croke and Ivanov, {Sergei V.} and Katz, {Mikhail G.}",
year = "2007",
month = jan,
day = "1",
doi = "10.1090/S0002-9947-06-03836-0",
language = "English",
volume = "359",
pages = "1--17",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "1",

}

RIS

TY - JOUR

T1 - Boundary case of equality in optimal loewner-type inequalities

AU - Bangert, Victor

AU - Croke, Christopher

AU - Ivanov, Sergei V.

AU - Katz, Mikhail G.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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 Tb, 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 Lp-norms. Here we exploit Lp-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 Tb, 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.

AB - 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 Tb, 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 Lp-norms. Here we exploit Lp-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 Tb, 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.

KW - Abel-Jacobi map

KW - Conformal systole

KW - Deformation theorem

KW - Extremal lattice

KW - Free abelian cover

KW - Generalized degree

KW - Isoperimetric inequality

KW - John ellipsoid

KW - Loewner inequality

KW - Lp-minimizing differential forms

KW - Perfect lattice

KW - Riemannian submersion

UR - http://www.scopus.com/inward/record.url?scp=34548532348&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-06-03836-0

DO - 10.1090/S0002-9947-06-03836-0

M3 - Article

AN - SCOPUS:34548532348

VL - 359

SP - 1

EP - 17

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -

ID: 50975596