Quasi-Fuchsian Co-Minkowski Manifolds

Since the work of W.P. Thurston, some maps from Teichmüller space into itself can be described using the extrinsic geometry of surfaces in three dimensional hyperbolic space. Similarly, since the work of G. Mess, some of these maps can be described using the extrinsic geometry of surfaces in Lorentzian space-forms. Here we will use the degenerate geometry of co-Minkowski space to prove, and generalize to any dimension, a theorem of Thurston that says that the total length of measured geodesic laminations provides an asymmetric norm on the tangent bundle of Teichmüller space. The main part of this chapter 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 construction. 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 then obtain an extrinsic proof of the theorem of Thurston mentioned above. We also exhibit and comment on the Anosov-like character of these deformations, similar to the Anosov character of the quasi-Fuchsian representations pointed out in Guichard and Wienhard (Invent Math 190(2):357–438, 2012).