The Morse landscape of a riemannian disk

S. Frankel; Michael Katz

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 2, page 503-507
  • ISSN: 0373-0956

Abstract

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We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real-valued function on the tree. Our construction answers in the negative a question of Gromov, which was motivated by the study of the word metric on finitely generated groups.

How to cite

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Frankel, S., and Katz, Michael. "The Morse landscape of a riemannian disk." Annales de l'institut Fourier 43.2 (1993): 503-507. <http://eudml.org/doc/75007>.

@article{Frankel1993,
abstract = {We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real-valued function on the tree. Our construction answers in the negative a question of Gromov, which was motivated by the study of the word metric on finitely generated groups.},
author = {Frankel, S., Katz, Michael},
journal = {Annales de l'institut Fourier},
keywords = {length functional; Riemannian disks; diameter; circumference; imbedded finite trees; complexity; word metric},
language = {eng},
number = {2},
pages = {503-507},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Morse landscape of a riemannian disk},
url = {http://eudml.org/doc/75007},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Frankel, S.
AU - Katz, Michael
TI - The Morse landscape of a riemannian disk
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 2
SP - 503
EP - 507
AB - We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real-valued function on the tree. Our construction answers in the negative a question of Gromov, which was motivated by the study of the word metric on finitely generated groups.
LA - eng
KW - length functional; Riemannian disks; diameter; circumference; imbedded finite trees; complexity; word metric
UR - http://eudml.org/doc/75007
ER -

References

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  1. [1] M. GROMOV, Asymptotic invariants of infinite groups, IHES preprint, 1992. 

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