Cargando…

The algebraic structure of geometric flows in two dimensions

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional a...

Descripción completa

Detalles Bibliográficos
Autor principal: Bakas, Ioannis
Lenguaje:eng
Publicado: 2005
Materias:
Acceso en línea:https://dx.doi.org/10.1088/1126-6708/2005/10/038
http://cds.cern.ch/record/860055
_version_ 1780907161883246592
author Bakas, Ioannis
author_facet Bakas, Ioannis
author_sort Bakas, Ioannis
collection CERN
description There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.
id cern-860055
institution Organización Europea para la Investigación Nuclear
language eng
publishDate 2005
record_format invenio
spelling cern-8600552023-03-14T18:48:09Zdoi:10.1088/1126-6708/2005/10/038http://cds.cern.ch/record/860055engBakas, IoannisThe algebraic structure of geometric flows in two dimensionsParticle Physics - TheoryThere is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.hep-th/0507284CERN-PH-TH-2005-134CERN-PH-TH-2005-134oai:cds.cern.ch:8600552005-07-28
spellingShingle Particle Physics - Theory
Bakas, Ioannis
The algebraic structure of geometric flows in two dimensions
title The algebraic structure of geometric flows in two dimensions
title_full The algebraic structure of geometric flows in two dimensions
title_fullStr The algebraic structure of geometric flows in two dimensions
title_full_unstemmed The algebraic structure of geometric flows in two dimensions
title_short The algebraic structure of geometric flows in two dimensions
title_sort algebraic structure of geometric flows in two dimensions
topic Particle Physics - Theory
url https://dx.doi.org/10.1088/1126-6708/2005/10/038
http://cds.cern.ch/record/860055
work_keys_str_mv AT bakasioannis thealgebraicstructureofgeometricflowsintwodimensions
AT bakasioannis algebraicstructureofgeometricflowsintwodimensions