Abstract
In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space for the non-compact symmetric space mathbb {RH}^n and show how SO_0(n,1)-Higgs bundles can be used to parametrise this space, making clear how the classical invariants (induced metric and second fundamental form) figure in this picture. We use this parametrisation to provide details of the moduli spaces for mathbb {RH}^3 and mathbb {RH}^4, and relate their structure to the structure of the corresponding Higgs bundle moduli spaces.
Highlights
In this article we study equivariant minimal immersions f : D → N of the Poincaré disc D into a non-compact symmetric space N
In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two
We show in Theorem 2.4 that this is the Higgs bundle for an equivariant minimal immersion if and only if it is polystable and V is constructed from an S O(n − 2, C) bundle (W, QW ) and a cohomology class ξ ∈ H 1(Hom(W, K −1)), where K is the canonical bundle of c, as follows
Summary
In this article we study equivariant minimal immersions f : D → N of the Poincaré disc D into a non-compact symmetric space N. The link to minimal surfaces is that the non-abelian Hodge correspondence already tells us about all equivariant (or “twisted”) harmonic maps: fix a Fuchsian representation c of π1 into the group Isom+(D) of oriented isometries of the Poincaré disc D, so that c D/c(π1 ). Labourie conjectured in [25] that every Hitchin representation into a split real form should admit a unique minimal surface, which would imply that the moduli space of Hitchin representations (whose parametrisation is due to Hitchin [22]) provides components of M( , N ) when the isometry group G of N is a split real form Labourie recently proved his conjecture for split real forms of rank-two complex simple Lie groups [26].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
