Jeffrey Danciger

A geometric transition from hyperbolic to anti-de Sitter geometry

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti-de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the “other side” of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the .2;m;m/ triangle orbifold for m 5.

Convex projective structures on nonhyperbolic three-manifolds

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoists theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

Ideal triangulations and geometric transitions

Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter (AdS) geometry. Our approach involves solving Thurstons gluing equations over several different shape parameter algebras. In the case of a punctured torus bundle with Anosov monodromy, we identify two components of real solutions for which there are always nearby positively oriented solutions over both the complex and pseudo-complex numbers. These complex and pseudo-complex solutions define hyperbolic and AdS structures that, after coordinate change in the projective model, may be arranged into one continuous family of real projective structures. We also study the rigidity properties of certain AdS structures with tachyon singularities.

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups

Geometry and topology of complete Lorentz spacetimes of constant curvature

\mathrm {PO}(p,q)

Variational Principles for Symmetric Bilinear Forms

PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space

Convex cocompact actions in real projective geometry

\mathbb {H}^{p,q-1}