WorkshopsMinimal Lagrangian surfaces in the complex hyperbolic plane and surface group representations
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Following the work of Hitchin, Donaldson and Corlette, the study of representations of a surface group (fundamental group of a closed Riemann surface) into SU(2,1) is related to the study of twisted harmonic maps of the surface into the complex hyperbolic plane. The space of (reductive) representations has finitely many connected components indexed by their Toledo invariant. John Loftin (Rutgers, Newark) and I have recently shown that one can construct an open subset in the component with Toledo invariant zero corresponding to twisted minimal Lagrangian immersions. The differential geometric data for such maps (the conformal structure and a holomorphic cubic differential) gives a parameterisation for this open set which makes clear the underlying role of Teichmuller space. This construction begs the question: how far can this parameterisation be extended (within the Toledo invariant zero component)? This is an open question and presents a challenge for the application of minimal Lagrangian submanifold theory. The talk will assume no background in representation theory.
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Ian McIntosh
2011-07-07
14:00:00 - 15:00:00
R101 , Astronomy and Mathematics Building
Following the work of Hitchin, Donaldson and Corlette, the study of representations of a surface group (fundamental group of a closed Riemann surface) into SU(2,1) is related to the study of twisted harmonic maps of the surface into the complex hyperbolic plane. The space of (reductive) representations has finitely many connected components indexed by their Toledo invariant. John Loftin (Rutgers, Newark) and I have recently shown that one can construct an open subset in the component with Toledo invariant zero corresponding to twisted minimal Lagrangian immersions. The differential geometric data for such maps (the conformal structure and a holomorphic cubic differential) gives a parameterisation for this open set which makes clear the underlying role of Teichmuller space. This construction begs the question: how far can this parameterisation be extended (within the Toledo invariant zero component)? This is an open question and presents a challenge for the application of minimal Lagrangian submanifold theory. The talk will assume no background in representation theory.
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