The Near Horizon Geometry equation on compact 2-manifolds including the general solution for g > 0

The Near Horizon Geometry (NHG) equation with a cosmological constant Λ 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 those points of a given solution where the component Ψ2 of the Weyl tensor does not vanish. Otherwise the Type D equation was not applicable. In the current paper we prove that the vanishing of Ψ2 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 Λ=0 case are generalized to arbitrary value of Λ. They may be still useful for compact 2-manifolds of positive Euler characteristic.