Almost-Fuchsian representations and holomorphic immersions in complex hyperbolic space

A representation is said to be almost-Fuchsian if it admits a nice equivariant map with strong geometric properties. In turn, it implies that the representation is convex-cocompact, hence discrete and faithful.

In this talk we will explain how to build almost-fuchsian representations in the group PU(2,1) with Toledo invariant close to being maximal. These will admit equivariant holomorphic almost-fuchsian immersions in the complex hyperbolic space.