Quasi-Fuchsian co-Minkowski manifolds

Abstract

This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there is a convex core. There is also a unique "mean" hypersurface, i.e. with traceless second fundamental form. The mean distance between the mean hypersurface and the lower boundary of the convex core endows the space of affine deformations of a given lattice with an asymmetric norm. The symmetrization of the asymmetric norm is simply the volume of the convex core. In dimension 2+1, the asymmetric norm is the total length of the bending lamination of the lower boundary component of the convex core. We obtain an extrinsic proof of a theorem of Thurston saying that, on the tangent space of Teichm\"{u}ller space, the total length of measured geodesic laminations is an asymmetric norm. We also exhibit and comment the Anosov-like character of these deformations, similar to the Anosov character of the quasi-Fuchsians representations pointed out in Guichard--Wienhard.