Let $\Gamma$ be a finitely generated group, and let $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$ be the moduli space of representations of $\Gamma$ into $\mathrm{SO}(2, n) (n \geq 2)$. An element $\rho : \Gamma \to \mathrm{SO}(2, n)$ of $\mathrm{Rep}(\Gamma , \mathrm{SO}(2, n))$ is quasi-Fuchsian if it is faithful, discrete, preserves an acausal $(n-1)$-sphere in the conformal boundary $\mathrm{Ein}_n$ of the anti-de Sitter space, and if the associated globally hyperbolic anti-de Sitter space is spatially compact—a particular case is the case of Fuchsian representations, i.e., composition of a faithful, discrete, and cocompact representation $\rho_f : \Gamma \to \mathrm{SO}(1, n)$ and the inclusion $\mathrm{SO}(1, n) \subset \mathrm{SO}(2, n)$.

In Anosov AdS representations are quasi-Fuchsian we proved that quasi-Fuchsian representations are precisely representations that are Anosov as defined in Anosov flows, surface groups and curves in projective space. In the present paper, we prove that the space of quasi-Fuchsian representations is open and closed, i.e., that it is a union of connected components of $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$.

The proof involves the following fundamental result: Let $\Gamma$ be the fundamental group of a globally hyperbolic spatially compact spacetime locally modeled on $\mathrm{AdS}_n$, and let $\rho : \Gamma \to SO_0(2, n)$ be the holonomy representation. Then, if $\Gamma$ is Gromov hyperbolic, the $\rho (\Gamma)$-invariant achronal limit set in $\mathrm{Ein}_n$ is acausal.

Finally, we also provide the following characterization of representations with zero-bounded Euler class: they are precisely the representations preserving a closed achronal subset of $\mathrm{Ein}_n$.

We prove that on a compact Sasakian manifold $(M,\eta, g)$ of dimension $2n + 1$, for any $0 \leq p \leq n$ the wedge product with $\eta \wedge (d \eta)^p$ defines an isomorphism between the spaces of harmonic forms $\Omega^{n-p}_{\Delta}(M)$ and $\Omega^{n+p+1}_{\Delta}(M)$. Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p}(M)$ and $H^{n+p+1}(M)$. Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1 \times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.

The centro-affine Minkowski problem in affine differential geometry is considered. Existence for the solution of the discrete centro-affine Minkowski problem is proved.