I am looking for reference books about real-valued harmonic functions on complete Riemannian surfaces, do you have any reference in your mind about this? I found some books about harmonic functions on the plane (but mainly discussing on conformal maps) or about general harmonic maps (which is too general to possess nice properties). They do not fit my need...
Precisely, I would like to understand the zero set (for example, is it a one dimensional manifold?), the existence and linearity of such functions.
Thanks for your help in advance!