Geometriae Dedicata最新文献_第8页

Cohomogeneity one solitons for the isometric flow of G 2 -structures. G 2 结构等距流的同质一孤子。

IF 0.5 4区数学

Geometriae Dedicata Pub Date : 2024-01-01 Epub Date: 2024-09-30 DOI: 10.1007/s10711-024-00954-8

Thomas A Ivey, Spiro Karigiannis

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Cohomogeneity one solitons for the isometric flow of <ns0:math><ns0:msub><ns0:mtext>G</ns0:mtext> <ns0:mn>2</ns0:mn></ns0:msub> </ns0:math> -structures.","authors":"Thomas A Ivey, Spiro Karigiannis","doi":"10.1007/s10711-024-00954-8","DOIUrl":"https://doi.org/10.1007/s10711-024-00954-8","url":null,"abstract":"We consider the existence of cohomogeneity one solitons for the isometric flow of <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -structures on the following classes of torsion-free <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -manifolds: the Euclidean <math> <msup><mrow><mi>R</mi></mrow> <mn>7</mn></msup> </math> with its standard <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on <math> <msup><mrow><mi>R</mi></mrow> <mn>7</mn></msup> </math> is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11442535/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142367550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Divergence of separated nets with respect to displacement equivalence. 分离网在位移等价方面的散度。

IF 0.5 4区数学

Geometriae Dedicata Pub Date : 2024-01-01 Epub Date: 2023-11-17 DOI: 10.1007/s10711-023-00862-3

Michael Dymond, Vojtěch Kaluža

{"title":"Divergence of separated nets with respect to displacement equivalence.","authors":"Michael Dymond, Vojtěch Kaluža","doi":"10.1007/s10711-023-00862-3","DOIUrl":"https://doi.org/10.1007/s10711-023-00862-3","url":null,"abstract":"We introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions <math><mrow><mi>ϕ</mi><mo>:</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math>. Two separated nets are called <math><mi>ϕ</mi></math>-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most <math><mrow><mi>ϕ</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>. We show that the spectrum of <math><mi>ϕ</mi></math>-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded <math><mi>ϕ</mi></math>, to the indiscrete equivalence relation, corresponding to <math><mrow><mi>ϕ</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>∈</mo><mi>Ω</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of <math><mi>ϕ</mi></math>-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of <math><mrow><mi>ϕ</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math> for <math><mrow><mi>R</mi><mo>→</mo><mi>∞</mi></mrow></math>. We further undertake a comparison of our notion of <math><mi>ϕ</mi></math>-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of <math><mi>ϕ</mi></math>-displacement equivalence with that of bilipschitz equivalence.","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10656347/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138464551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Knot groups, quandle extensions and orderability 结群、Qandle 扩展和有序性

IF 0.5 4区数学

Geometriae Dedicata Pub Date : 2023-12-29 DOI: 10.1007/s10711-023-00876-x

Idrissa Ba, Mohamed Elhamdadi