Hyperbolic ends with particles and grafting on singular surfaces

Abstract

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by "smooth grafting". Theorem 1.5. Let M ∈ HE θ be a non-degenerate hyperbolic end with particles, and let M d ∈ DS θ be the dual future-complete convex GHM de Sitter spacetime with particles. Given a closed, strictly concave surface S ⊂ M , there is a unique strictly future-convex spacelike surface S d and a unique diffeomorphism u : S → S d such that u * I d = III and u * III d = I, where I, III are the induced metric and third fundamental form on S, and I d and III d are the induced metric and third fundamental form on S d .Conversely, given any space-like, strictly future-convex S d surface in M d , there is a unique strictly concave surface S in M such that S d is the dual of S in the sense of Theorem 1.5. Proposition 1.6. Let S be a strictly concave surface in M , and let S d be the dual surface in M d . Then S has constant curvature K ∈ (−1, 0) if and only if S d has constant curvature K d = K/(K + 1) ∈ (−∞, 0).

Properness for circle packings and Delaunay circle patterns on complex projective structures

“…It is important to note that these statements are only known to be true when working on closed surfaces or, for complete surfaces of finite area, with measured laminations that do not enter the cusps. Extensions to hyperbolic surfaces with cone singularities of angle less than π are also possible, see [9].…”