A journey through harmonic functions

Abstract

In the present work we concentrate on the study of harmonic functions on Riemannian manifolds. For this purpose we carry out a study of them starting with the classical Liouville theorem in the complex plane through its generalization in the Euclidean space and finally on Riemannian manifolds using Yau's gradient estimate. From there we find a method to understand the asymptotic behavior of harmonic functions on unbounded rotationally symmetric surfaces and, moreover, what consequences it has on the existence of non-constant bounded harmonic functions on this type of surfaces. Finally we concentrate on showing the duality that exists between the study of the growth of asymptotic functions and the Dirichlet problem at infinity.