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Title:
Integrability meets Control Theory: Harmonic Maps in General Relativity
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Abstract:
We provide a framework for analyzing axially symmetric harmonic maps on R3 with symmetric
target spaces G=K. Drawing on results from analysis to Lie theory to geometry, we give a complete
and rigorous proof that, all such maps are completely integrable. We further demonstrate that
new solutions to the harmonic map equations can be generated from a given seed solution, using a
dressing or vesture method. This unies the integrability of theories including chiral eld models,
nonlinear -models, Yang-Mills and Einstein electrovacuum equations in the general context of
harmonic maps.
Utility of the vesture method is made concrete by generating N-solitonic harmonic maps into a
noncompact Grassmann manifold G = SU(p; q). We demonstrate a special case by deriving Kerr
and Kerr-Newman solutions from the Minkowski initial seed for the Einstein vacuum and Einstein-
Maxwell cases, respectively. In performing an asymptotic analysis, these solutions are shown to be
in the hyperextreme sector of the corresponding parameters, suggesting constraints on the dressing
mechanism. We indicate the possibility of using this analysis to control the resulting N-black hole
configurations in this setting.
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