Quasi-Fuchsian Co-Minkowski Manifolds

Geometric transition from hyperbolic to anti-de Sitter structures in dimension four

“…Although this fact has already been observed, for instance in [FS19] and [BF18], we provide a complete proof since the explicit computation of the isomorphism will be useful in the remainder of the paper.…”

“…It is worth remarking that the cone-manifolds of Theorem 1.1 are non-compact, but of finite volume. (See [FS19, Chapter 5] and [BF18] for the notion of volume in half-pipe geometry.) Nevertheless, the singularity Σ is compact, or in other words, it does not enter into the ends of the cone-manifolds.…”

Measured foliations at infinity of quasi-Fuchsian manifolds near the Fuchsian locus

“…Now we consider quasi-Fuchsian half-pipe manifolds (see Definition 7.1) following [10], see also [1]. These are intermediary geometric structures which arise naturally when we consider a smooth transition between hyperbolic and anti de-Sitter structures on M via the Fuchsian locus.…”

“…Since ∂ ∞ H 3 is identified with CP 1 and ρ QF (π 1 (S)) acts by elements of P SL 2 (C), i.e, Möbius transformations, the components of the boundary at infinity of a quasi-Fuchsian manifold carry canonical complex projective structures or CP 1 -structures. Following, for example the survey in [12], a CP 1 -structure on S is an open covering of S by an atlas (U α , φ α ) such that φ α : U α → CP 1 are charts to open domains in CP 1 and in the overlap U i ∩ U j of two charts the change of coordinate map φ i • φ −1 j is locally restriction of Möbius transformations. Denote the space of equivalence classes of CP 1 -structures on S under diffeomorphisms isotopic to the identity as CP(S).…”

“…The Schwarzian derivative yields a parametrisation of the fibers of the forgetful map CP(S) → T (S), i.e., CP 1 -structures on S with the same underlying complex structure. Given the unit disc ∆ in C and u : ∆ → C a locally injective holomorphic map, the Schwarzian derivative σ(u) of u is a holomorphic quadratic differential defined locally on ∆ as ( §3.1, [12]):…”