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The adiabatic limit of wave map flow on a two-torus


Author: J. M. Speight
Journal: Trans. Amer. Math. Soc. 367 (2015), 8997-9026
MSC (2010): Primary 58Z05, 58J90
DOI: https://doi.org/10.1090/tran/6538
Published electronically: April 8, 2015
MathSciNet review: 3403078
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Abstract: The $ S^2$ valued wave map flow on a Lorentzian domain $ \mathbb{R}\times \Sigma $, where $ \Sigma $ is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps $ \Sigma \rightarrow S^2$ is considered, in the limit of small initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the $ L^2$ metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.


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Additional Information

J. M. Speight
Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, England
Email: speight@maths.leeds.ac.uk

DOI: https://doi.org/10.1090/tran/6538
Received by editor(s): May 1, 2013
Received by editor(s) in revised form: July 8, 2014
Published electronically: April 8, 2015
Article copyright: © Copyright 2015 American Mathematical Society