Thurston's asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

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dc.contributor.authorSaglam, Ismail
dc.contributor.authorPapadopoulos, Athanase
dc.date.accessioned2025-01-06T17:44:35Z
dc.date.available2025-01-06T17:44:35Z
dc.date.issued2024
dc.description.abstractConsider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface with a Euclidean rectangle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston's asymmetric metric on the Teichm & uuml;ller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichm & uuml;ller space to be geodesic and we prove that any two points of this space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm on the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the subspace topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.
dc.description.sponsorshipTUEBIdot;TAK [221N171]; CNRS-TUEBIdot;TAK; Teichmueller Theory of Hyperbolicand Flat Surfaces with Conical Singularities [221N171]
dc.description.sponsorshipThe first named author is financially supported by TUB & Idot;TAK. Both authors are supported by CNRS-TUB & Idot;TAK joint project Teichmuller Theory of Hyperbolic and Flat Surfaces with Conical Singularities No. 221N171.
dc.identifier.doi10.4171/LEM/1060
dc.identifier.endpage250
dc.identifier.issn0013-8584
dc.identifier.issn2309-4672
dc.identifier.issue1-2
dc.identifier.startpage231
dc.identifier.urihttps://doi.org/10.4171/LEM/1060
dc.identifier.urihttps://hdl.handle.net/20.500.14669/3110
dc.identifier.volume70
dc.identifier.wosWOS:001229882700009
dc.identifier.wosqualityN/A
dc.indekslendigikaynakWeb of Science
dc.language.isoen
dc.publisherEuropean Mathematical Soc-Ems
dc.relation.ispartofEnseignement Mathematique
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı
dc.rightsinfo:eu-repo/semantics/openAccess
dc.snmzKA_20241211
dc.subjectThurston's asymmetric metric
dc.subjectTeichm & uuml;ller theory
dc.subjectgeodesics
dc.subjectFinsler structure
dc.titleThurston's asymmetric metric on the space of singular flat metrics with a fixed quadrangulation
dc.typeArticle

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