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Almost-Fuchsian representations in PU(2,1)

Samuel Bronstein

TL;DR

The work advances the construction of almost-Fuchsian representations into PU$(2,1)$ with nonmaximal Toledo invariant by developing a robust analytic framework for holomorphic equivariant immersions, combining curvature PDEs, Higgs-bundle stability, and fixed-point methods. Central to the approach is solving a coupled system for $(u,v)$ and a holomorphic section $eta$ to produce holomorphic immersions whose second fundamental form is uniformly small, yielding convex-cocompact representations with Toledo invariants of the form $Tol( ho)=2-2g+ rac{2}{3}d$. For large genus, the authors produce representations that do not lift to SU$(2,1)$ when $d otigm|3$, thereby answering Loftin–McIntosh’s question in the PU$(2,1)$ setting and enriching the landscape of noninteger Toledo invariants. The method generalizes the degree-1 disc-bundle constructions known in higher-dimensional hyperbolic geometries and provides a fixed-point framework tied to balanced line bundles, offering a nonasymptotic route to almost-Fuchsian holomorphic maps. Overall, the paper deepens the link between holomorphic data, Higgs-bundle stability, and convex-cocompact representations in complex hyperbolic geometry, with explicit Toledo-invariant realizations and lifting obstructions.

Abstract

In this paper, we study nonmaximal representations of surface groups in PU(2,1). In genus large enough, we show the existence of convex-cocompact representations of non-maximal Toledo invariant admitting a unique equivariant minimal surface, which is holomorphic and almost totally geodesic. These examples can be obtained for any Toledo invariant of the form 2-2g +2/3 d, provided g is large compared to d. When d is not divisible by 3, this yields examples of convex-cocompact representations in PU(2,1) which do not lift to SU(2,1)

Almost-Fuchsian representations in PU(2,1)

TL;DR

The work advances the construction of almost-Fuchsian representations into PU with nonmaximal Toledo invariant by developing a robust analytic framework for holomorphic equivariant immersions, combining curvature PDEs, Higgs-bundle stability, and fixed-point methods. Central to the approach is solving a coupled system for and a holomorphic section to produce holomorphic immersions whose second fundamental form is uniformly small, yielding convex-cocompact representations with Toledo invariants of the form . For large genus, the authors produce representations that do not lift to SU when , thereby answering Loftin–McIntosh’s question in the PU setting and enriching the landscape of noninteger Toledo invariants. The method generalizes the degree-1 disc-bundle constructions known in higher-dimensional hyperbolic geometries and provides a fixed-point framework tied to balanced line bundles, offering a nonasymptotic route to almost-Fuchsian holomorphic maps. Overall, the paper deepens the link between holomorphic data, Higgs-bundle stability, and convex-cocompact representations in complex hyperbolic geometry, with explicit Toledo-invariant realizations and lifting obstructions.

Abstract

In this paper, we study nonmaximal representations of surface groups in PU(2,1). In genus large enough, we show the existence of convex-cocompact representations of non-maximal Toledo invariant admitting a unique equivariant minimal surface, which is holomorphic and almost totally geodesic. These examples can be obtained for any Toledo invariant of the form 2-2g +2/3 d, provided g is large compared to d. When d is not divisible by 3, this yields examples of convex-cocompact representations in PU(2,1) which do not lift to SU(2,1)

Paper Structure

This paper contains 18 sections, 27 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Comparison with the case of $\mathrm{SO}(4,1)$
  3. Some Context: Complex hyperbolic geometry and almost-Fuchsian representations
  4. convex-cocompact representations in $\mathrm{PU}(2,1)$
  5. almost-Fuchsian representations
  6. holomorphic equivariant immersions in $\mathrm{PU}(2,1)$
  7. Acknowledgments
  8. Holomorphic immersions in the complex hyperbolic plane
  9. The curvature equations
  10. Stability conditions from the associated Higgs bundle
  11. Almost-Fuchsian immersions in the complex hyperbolic plane
  12. Constraints on almost-fuchsian representations
  13. The curvature equations of holomorphic immersions
  14. The Gauss Equation of a holomorphic immersion
  15. Study of the Poisson Equation
  16. ...and 3 more sections

Key Result

Theorem 1

Let $d>0$ and $\eta>0$. There is a genus $g_0$ such that for every closed surface of genus $g>g_0$, there exists a representation $\rho:\pi_1(\Sigma_g)\rightarrow\mathrm{PU}(2,1)$ almost-fuchsian, admitting an equivariant holomorphic map $f$, verifying:

Theorems & Definitions (50)

  • Theorem : A
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2.1
  • proof
  • Remark : Comparison with the Higgs bundles parametrization
  • Proposition 2.2
  • proof
  • Remark
  • Definition 3.1
  • ...and 40 more