What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions (locally) on 2-dimensional Riemannian manifolds, but this seems to be non-trivial, too. I want to have a construction using 1-dimensional complex analysis and not much else.

A similar question was asked here.

What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions (locally) on 2-dimensional Riemannian manifolds, but this seems to be non-trivial, too. I want to have a construction using 1-dimensional complex analysis and not much else.

A similar question was asked here.