Maps from a surface into a compact Lie group and curvature

  • Andres Larrain-HubachEmail author
  • Doug Pickrell


This is a brief note on curvature properties of Sobolev Lie groups of maps from a Riemann surface into a compact Lie group K. Freed showed that, in a necessarily qualified sense, the quotient space \(W^{1/2}(S^1,K)/K\) is a (nonnegative constant) Einstein “manifold” with respect to the essentially unique \({\hbox {PSU}}(1,1)\)-invariant metric, where \(W^{s}\) denotes maps of \(L^2\) Sobolev order s. In a similarly qualified sense, and in addition making use of the Dixmier trace/Wodzicki residue, we show that for a Riemann surface \(\Sigma \), \(W^1(\Sigma ,K)/K\) is a (nonnegative constant) Einstein “manifold” with respect to the essentially unique conformally invariant metric. Because of the qualifications involved in these statements, in practice it is necessary to consider curvature for \(W^s(\Sigma ,K)\) for s above the critical exponent, and limits.


Infinite dimensional Lie groups Pseudo differential operators Wodzicki residue 

Mathematics Subject Classification

22E65 22E70 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.University of DaytonDaytonUSA
  2. 2.Mathematics departmentUniversity of ArizonaTucsonUSA

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