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Harmonic maps for Hitchin representations

  • Qiongling LiEmail author
Article
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Abstract

Let (S,g0) be a hyperbolic surface, \({\rho}\) be a Hitchin representation for \({PSL(n,{\mathbb{R}})}\), and f be the unique \({\rho}\)-equivariant harmonic map from \({({\widetilde{S}}, \widetilde g_0)}\) to the corresponding symmetric space. We show its energy density satisfies \({e(f)\geq 1}\) and equality holds at one point only if \({e(f)\equiv 1}\) and \({\rho}\) is the base \({n}\)-Fuchsian representation of (S,g0). In particular, we show given a Hitchin representation \({\rho}\) for \({PSL(n,{\mathbb{R}})}\), every \({\rho}\)-equivariant minimal immersion f from the hyperbolic plane \({{\mathbb{H}}^2}\) into the corresponding symmetric space X is distance-increasing, i.e. \({f^*g_{X}\geq g_{{\mathbb{H}}^2}}\). Equality holds at one point only if it holds everywhere and \({\rho}\) is an n-Fuchsian representation.

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Notes

Acknowledgements

The author wants to thank the referee for many useful comments and corrections. The author is supported in part by the center of excellence Grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95). The author acknowledges support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). The author acknowledges support from Nankai Zhide Foundation.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina
  2. 2.Centre for Quantum Geometry of Moduli Spaces (QGM), Department of MathematicsAarhus UniversityAarhus CDenmark
  3. 3.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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