Classes of harmonic functions in 2D generalized Poincare geometry

Abstract

By using the additive and multiplicative separation of variables we find some classes of solutions of the Laplace equation for a generalization of the Poincare upper half plane metric. Non-constant totally geodesic functions implies the at metric and several examples are studied including the Hamilton's cigar Ricci soliton. The Bochner formula is discussed for our generalized Poincare metric and for its important particular cases.