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RICCI FLOWS ON SURFACES RELATED TO THE EINSTEIN WEYL AND ABELIAN VORTEX EQUATIONS

Published online by Cambridge University Press:  22 August 2014

DANIEL J. F. FOX
DANIEL J. F. FOX*
Affiliation:
Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid España e-mail: [email protected]
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Abstract

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There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014 , pp. 569 - 599
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

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