The Near Horizon Geometry Equation on Compact 2-Manifolds Including the General Solution for g > 0

TL;DR

This paper classifies all solutions to the Near Horizon Geometry equation with a cosmological constant on compact 2-manifolds, showing solutions satisfy the Type D equation everywhere and extending previous results to degenerate points.

Contribution

It proves that degeneracy is eliminated by compactness and finds all solutions on manifolds with non-positive Euler characteristic, generalizing earlier integrability conditions.

Findings

Solutions satisfy the Type D equation at every point.

Degeneracy is ruled out by compactness.

Complete solutions are found for manifolds with non-positive Euler characteristic.

Abstract

The Near Horizon Geometry (NHG) equation with a cosmological constant {\Lambda} is considered on compact 2-dimensional manifolds. It is shown that every solution satisfies the Type D equation at every point of the manifold. A similar result known in the literature was valid only for non-degenerate in a suitable way points of a given solution. At the degenerate points the Type D equation was not applicable. In the current paper we prove that the degeneracy is ruled out by the compactness. Using that result we find all the solutions to the NHG equation on compact 2-dimensional manifolds of non-positive Euler characteristics. Some integrability conditions known earlier in the {\Lambda} = 0 case are generalized to arbitrary value of {\Lambda}. They may be still useful for compact 2-manifolds of positive Euler characteristic.