Abstract:
We consider the discrete representations of 3-manifold groups into $PU(2,1)$ that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. This illustrates the fact that a discrete representation $\pi_1(M)\rightarrow PU(2,1)$ with cyclic unipotent boundary holonomy is not in general the holonomy of a spherical CR uniformization of $M$.

Abstract:
In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms of its holonomy representation.

Abstract:
Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.

Abstract:
We characteristize those Einstein four manifolds which are locally symmetric spaces of noncompact type. Namely they are four manifolds which admit solutions to the (non-Abelian) Seiberg Witten equations and satisty certain characterisitc number equality.

Abstract:
We describe a simple fundamental domain for the holonomy group of the Deraux-Falbel spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the Deraux-Falbel holonomy group such that the boundary holonomy is parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-conjugate spherical CR uniformizations of the figure eight knot complement.

Abstract:
Consider a three dimensional cusped spherical $\mathrm{CR}$ manifold $M$ and suppose that the holonomy representation of $\pi_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{CR}$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz's spherical $\mathrm{CR}$ Dehn surgery theorem, but has weaker hypotheses and does not give the unifomizability of the structure. We apply our theorem in the case of the Deraux-Falbel structure on the Figure Eight knot complement and obtain spherical $\mathrm{CR}$ structures on all Dehn surgeries of slope $-3 + r$ for $r \in \mathbb{Q}^{+}$ small enough.

Abstract:
We study the action of a real-reductive group $G=K\exp(\lie{p})$ on real-analytic submanifold $X$ of a K\"ahler manifold $Z$. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^\mbb{C}$ such that the action of a maximal Hamiltonian subgroup is Hamiltonian. The moment map $\mu$ induces a gradient map $\mu_\lie{p}\colon X\to\lie{p}$. We show that $\mu_\lie{p}$ almost separates the $K$--orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion's characterization of spherical K\"ahler manifolds with moment maps.