Hyperbolic ends with particles and grafting on singular surfaces

TL;DR

This paper establishes the existence and uniqueness of constant Gauss curvature foliations in hyperbolic ends with particles and convex GHM de Sitter spacetimes with particles, linking geometric structures via duality and proving a homeomorphism for grafting maps.

Contribution

It introduces a novel duality between hyperbolic ends with particles and convex GHM de Sitter spacetimes, and proves the homeomorphism of grafting maps for surfaces with cone singularities.

Findings

Unique foliation by constant Gauss curvature surfaces in hyperbolic ends with particles.

Homeomorphism of grafting maps for surfaces with cone singularities.

Duality between hyperbolic ends with particles and de Sitter spacetimes.

Abstract

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $\pi$) 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 Teichm\"uller 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 $\pi$, as well as an analogue when grafting is replaced by "smooth grafting".