Gluing-at-infinity of two-dimensional asymptotically locally hyperbolic manifolds

We review notions of mass of asymptotically locally Anti-de Sitter three-dimensional spacetimes, and apply them to some known solutions. For two-dimensional general relativistic initial data sets the mass is not invariant under asymptotic symmetries, but a unique mass parameter can be obtained either by minimisation, or by a monodromy construction, or both. We give an elementary proof of positivity, and of a Penrose-type inequality, in a natural gauge. We apply the "Maskit gluing construction" to time-symmetric asymptotically locally hyperbolic vacuum initial data sets and derive mass/entropy formulae for the resulting manifolds. Finally, we show that all mass aspect functions can be realised by constant scalar curvature metrics on complete manifolds which are smooth except for at most one conical singularity.