Diastolic and isoperimetric inequalities on surfaces

Florent Balacheff; Stéphane Sabourau

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 4, page 579-605
  • ISSN: 0012-9593

Abstract

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We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger’s constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.

How to cite

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Balacheff, Florent, and Sabourau, Stéphane. "Diastolic and isoperimetric inequalities on surfaces." Annales scientifiques de l'École Normale Supérieure 43.4 (2010): 579-605. <http://eudml.org/doc/272169>.

@article{Balacheff2010,
abstract = {We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger’s constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.},
author = {Balacheff, Florent, Sabourau, Stéphane},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Cheeger constant; closed geodesics; curvature-free inequalities; diastole; isoperimetric inequalities; one-cycles},
language = {eng},
number = {4},
pages = {579-605},
publisher = {Société mathématique de France},
title = {Diastolic and isoperimetric inequalities on surfaces},
url = {http://eudml.org/doc/272169},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Balacheff, Florent
AU - Sabourau, Stéphane
TI - Diastolic and isoperimetric inequalities on surfaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 4
SP - 579
EP - 605
AB - We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger’s constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.
LA - eng
KW - Cheeger constant; closed geodesics; curvature-free inequalities; diastole; isoperimetric inequalities; one-cycles
UR - http://eudml.org/doc/272169
ER -

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