Almost-Fuchsian representations in PU(2,1)

TL;DR

This paper explores nonmaximal surface group representations in PU(2,1), demonstrating the existence of convex-cocompact examples with unique minimal surfaces and specific Toledo invariants, including cases not liftable to SU(2,1).

Contribution

It introduces new convex-cocompact representations with unique minimal surfaces for large genus and specific Toledo invariants, expanding understanding of surface group representations in PU(2,1).

Findings

Existence of convex-cocompact representations with unique minimal surfaces.

Construction of examples for Toledo invariants of the form 2-2g + 2/3 d.

Identification of representations that do not lift to SU(2,1).

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)