Isosystolic genus three surfaces critical for slow metric variations
type: Item
instance of: scholarly article
title: Isosystolic genus three surfaces critical for slow metric variations
zbMATH Open document ID: 1246.53061
author: Stéphane Sabourau
published in: Geometry \& Topology
publication date: 2011-08-19T00:00:00Z
review text: Let \(M\) be a non-simply-connected closed surface. The systolic area of a Riemannian metric \(g\) on \(M\) is \(\sigma(g)=\mathrm{area}\,(g) / \mathrm{sys}\,(g)^2\), where \(\mathrm{sys}\,(g)\) is the infimum of the lengths of noncontractible loops on \(M\). The metric \(g\) is extremal if it realizes the minimum of the systolic area for a given topological surface. Very few such metrics are known at present, and none on the surfaces of genus higher than \(1\). E. Calabi described two highly symmetric piecewise flat metrics on surfaces of genus 3 and conjectured that one of them is extremal [\textit{E. Calabi}, in: Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Luminy, France, 12--18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 167--204 (1996; Zbl 0884.58029)]. The author provides some evidence in support of this conjecture by showing that both metrics are critical with respect to variations of certain type, called slow metric variations. The slowness condition is defined in terms of the differential of the Busemann function. It remains unknown whether this condition is satisfied for all metric variations.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C38
zbMATH DE Number: 5940697
zbMATH Keywords: systole
zbMATH Keywords: systolic inequality
zbMATH Keywords: extremal metric
zbMATH Keywords: Calabi surface
zbMATH Keywords: Busemann function
zbMATH Keywords: metric variation
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
full work available at URL: gt.2011.15.1477
OpenAlex ID: W2092604837
cites work: Q3006857
cites work: On the systole of the sphere in the proximity of the standard metric
cites work: A LOCAL OPTIMAL DIASTOLIC INEQUALITY ON THE TWO-SPHERE
cites work: Inégalité isosystolique pour la bouteille de Klein. (Isosystolic inequalities for the Klein bottle)
cites work: Inégalités isosystoliques conformes pour la bouteille de Klein. (Conformal isosystolic inequalities for the Klein bottle)
cites work: Isosystolic conformal inequalities
cites work: Entropy and rigidity of locally symmetric spaces of strictly negative curvature
cites work: On extremals with prescribed Lagrangian densities
cites work: On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume.
cites work: Boundary rigidity and filling volume minimality of metrics close to a flat one
cites work: Q4340763
cites work: Q4463916
cites work: Deformations of metrics and associated harmonic maps
cites work: Riemannian manifolds admitting isometric immersions by their first eigenfunctions
cites work: Filling Riemannian manifolds
cites work: Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
cites work: Loewner's torus inequality with isosystolic defect
cites work: On two-dimensional minimal fillings
cites work: Entropy of systolically extremal surfaces and asymptotic bounds
cites work: Hyperelliptic surfaces are Loewner
cites work: An optimal systolic inequality for \(\mathrm{CAT}(0)\) metrics in genus two
cites work: Berger's isoperimetric problem and minimal immersions of surfaces
cites work: Some inequalities in certain nonorientable Riemannian manifolds
cites work: Systols of singular plane surfaces of genus two
cites work: Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic
cites work: Prescribing Curvature on Compact Surfaces with Conical Singularities
Wikidata QID: Q126190585
DOI: 10.2140/GT.2011.15.1477
Recommended article: Q4340763
Recommended article: Systols of singular plane surfaces of genus two
Recommended article: A systolic-like extremal genus two surface
Recommended article: Systolically extremal nonpositively curved surfaces are flat with finitely many singularities
Recommended article: On two dimensional isosystolic inequalities
Recommended article: On the systoliy geometry of Bieberbach manifolds
Recommended article: New proof of a Calabi's theorem
Recommended article: An optimal systolic inequality for \(\mathrm{CAT}(0)\) metrics in genus two
Recommended article: A systolic inequality for geodesic flows on the two-sphere
Recommended article: Asymptotic bounds for separating systoles on surfaces
description: scientific article; zbMATH DE number 5940697