Mixed Solutions to the Liouville Equation

TL;DR

This paper extends explicit solutions to the Liouville equation with mixed hyperbolic and elliptic monodromies, analyzing their limits and geometric interpretations related to black hole horizons in 3D gravity.

Contribution

It introduces new explicit solutions to the Liouville equation with mixed monodromies and explores their geometric and physical implications in three-dimensional gravity.

Findings

Explicit solutions with mixed monodromies are constructed.

Large hyperbolic monodromy limits are analyzed.

Geometric interpretations relate to black hole horizons.

Abstract

We enlarge the set of explicit classical solutions to the Liouville equation with three singularities to the cases with mixed hyperbolic and elliptic monodromies. We analyze the large hyperbolic monodromy limit of the solutions and the farthest geodesics looping one hyperbolic singularity. These two-dimensional geometries describe a time-symmetric spatial slice of a solution to three-dimensional general relativity. The geodesics are reinterpreted as snapshots of horizons of evolving black holes. We study the spatial slice with three horizons of very heavy black holes in some detail. We use uniform saddle point integration to present the Liouville and heavy black hole geometries in terms of simpler special functions. These make a detailed analysis of mixed particle and black hole geometries possible.