Abstract
This paper gives a construction for all minimal immersions ƒ of the Poincaré disc into the complex hyperbolic plane CH2 which are equivariant with respect to an irreducible representation ρ of a hyperbolic surface group into PU(2,1). We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs (ρ,ƒ) is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for PU(2,1), and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.
| Original language | English |
|---|---|
| Pages (from-to) | 71-106 |
| Number of pages | 37 |
| Journal | Asian Journal of Mathematics |
| Volume | 23 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 3 May 2019 |
Bibliographical note
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.Keywords
- math.DG
- 20H10, 53C43, 58E20
- Higgs bundle
- Complex hyperbolic plane
- Minimal surface
