Almost-Fuchsian representations in PU(2,1)

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)

💡 Research Summary

The paper investigates non‑maximal surface group representations into the isometry group PU(2,1) of complex hyperbolic two‑space. While maximal representations (those attaining the Milnor–Wood bound |τ|=2g‑2) are well‑understood and preserve a totally geodesic copy of the complex line, the landscape of non‑maximal components has remained largely unexplored. The author proves that for any positive integer d and any small constant η>0, there exists a genus g₀ such that for every closed oriented surface Σ_g with g>g₀ one can construct a convex‑cocompact representation
ρ:π₁(Σ_g)→PU(2,1)
with Toledo invariant
τ(ρ)=2−2g+ (2/3)d,
which is “almost‑Fuchsian”: it admits a unique ρ‑equivariant holomorphic minimal immersion f:˜Σ_g→H²_ℂ whose second fundamental form satisfies sup‖II_f‖<η. The construction works for any d, and when d is not divisible by 3 the Toledo invariant is non‑integral, implying that ρ does not lift to SU(2,1). This answers positively a question raised by Loftin–McIntosh about the existence of convex‑cocompact PU(2,1) representations that fail to lift.

The technical heart of the work is a system of coupled nonlinear PDEs (the “holomorphic Gauss–Codazzi equations”) for a pair of real functions u, v on the universal cover and a holomorphic cubic differential β∈H⁰(S,K³L⁻¹), where L is a holomorphic line bundle of degree d. Solving
Δu = 2e^{2u}−1 + e^{−4u−2v}‖β‖²,
Δv = λ−3e^{2u},
with λ determined by deg(L), yields a projectively flat PU(2,1) bundle equipped with a Hermitian metric of signature (2,1). The associated developing map is precisely the desired holomorphic immersion f. The (2,0) part of the second fundamental form of f is identified with β, so by choosing β sufficiently small one can make the norm of II_f arbitrarily small, achieving the almost‑Fuchsian condition.

Stability of the underlying Higgs bundle (E,∂,Φ) forces 0<deg(L)<3g−3, which is automatically satisfied when g is large compared to d. This stability guarantees the existence of the harmonic metric solving the PDE system and ensures that the flat PU(2,1) connection truly comes from a representation rather than merely a projective one.

Two immediate corollaries follow. First, for large genus there are convex‑cocompact PU(2,1) representations that do not lift to SU(2,1), providing the first examples of non‑integral Toledo invariant in the convex‑cocompact setting. Second, for any η>0 one obtains holomorphic embeddings of the complex line into H²_ℂ with ‖II_f‖<η but which are not totally geodesic. The boundary of the image is a quasicircle whose Hausdorff dimension can be made arbitrarily close to 1 by taking η small, illustrating that holomorphic minimal surfaces can be “almost flat” without being flat.

The paper also compares the situation with the real hyperbolic case SO(4,1). Earlier work produced almost‑Fuchsian representations corresponding to degree‑1 disc bundles; here the author works directly with degree‑d disc bundles, achieving a more synthetic construction that does not rely on Moser–Trudinger inequalities. An appendix shows that the same methods apply to H⁴, producing almost‑Fuchsian super‑minimal immersions whose quotients are diffeomorphic to degree‑d disc bundles over the surface.

In summary, the work combines complex hyperbolic geometry, Higgs bundle theory, and nonlinear analysis to exhibit a rich new family of convex‑cocompact, almost‑Fuchsian representations in PU(2,1) with non‑maximal, often non‑integral Toledo invariants. It expands the known landscape of surface group representations into complex hyperbolic groups, provides explicit geometric models for the associated manifolds, and opens new directions for studying minimal surfaces, deformation theory, and lifting problems in higher rank Hermitian symmetric spaces.