Maximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces

2015/03/11 by B. Collier, Brian Collier, Collier, Brian
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.DG #math.GT

paper · pdf · doi:10.48550/arxiv.1503.03526

37 pages, comments welcome, v2 mistakes in the proof Theorem 3.8 corrected

openalex publication_date 2015/03/11 · arxiv created 2015/07/06 · arxiv updated 2015/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a closed surface of genus at least 2. For each maximal representation ρ: π₁(S)\rightarrowSp(4,ℝ) in one of the 2g-3 exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space Sp(4,ℝ)/U(2) is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on Hitchin components, these bundles are not vector bundles.

Citations

• The Self-Duality Equations on a Riemann Surface
• G2 geometry and integrable systems
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Representations of the fundamental group of a closed oriented surface in Sp(4,R)
• High energy harmonic maps and degeneration of minimal surfaces
• Maximal Representations of Surface Groups: Symplectic Anosov Structures
• Hitchin harmonic maps are immersions
• Components of spaces of surface group representations