Geometric Cone Surfaces and (2+1)- Gravity coupled to Particles

TL;DR

This paper studies (2+1)-dimensional spacetimes with particles, constructed via cone surfaces and deformations, revealing their geometric properties and parameter space structure, especially under large mass conditions.

Contribution

It introduces new constructions of (2+1)-spacetimes using cone surfaces and deformations, analyzing their parameter space and topological properties.

Findings

Distinguished deformations form a large open subset of the parameter space.

The parameter space is homeomorphic to a product involving Teichmüller space.

Examples of spacetimes not arising from hyperbolic suspensions are constructed.

Abstract

We introduce the (2+1)-spacetimes with compact space of genus g and with r gravitating particles which arise by ``Minkowskian suspensions of flat or hyperbolic cone surfaces'', by ``distinguished deformations'' of hyperbolic suspensions and by ``patchworking'' of suspensions. Similarly to the matter-free case, these spacetimes have nice properties with respect to the canonical Cosmological Time Function. When the values of the masses are sufficiently large and the cone points are suitably spaced, the distinguished deformations of hyperbolic suspensions determine a relevant open subset of the full parameter space; this open subset is homeomorphic to the product of an Euclidean space of dimension 6g-6+2r with an open subset of the Teichm\"uller Space of Riemann surfaces of genus g with r punctures. By patchworking of suspensions one can produce examples of spacetimes which are not distinguished deformations of any hyperbolic suspensions, although they have the same masses; in fact, we will guess that they belong to different connected components of the parameter space.