On conjugate pseudo-harmonic functions

TL;DR

This paper establishes a necessary and sufficient condition for a real-valued continuous function to be a conjugate pseudo-harmonic function of a given pseudo-harmonic function on a surface, based on openness on level sets.

Contribution

It provides a characterization of conjugate pseudo-harmonic functions on surfaces through the openness condition on level sets.

Findings

Conjugate pseudo-harmonic functions are characterized by openness on level sets.

Theorem provides necessary and sufficient conditions for conjugacy.

Advances understanding of harmonic function conjugates on surfaces.

Abstract

We prove the following theorem. Let $U$ be a pseudo-harmonic function on a surface $M^2$. For a real valued continuous function $V : M^2 \to {\mathbb R}$ to be a conjugate pseudo-harmonic function of $U$ on $M^2$ it is necessary and sufficient that $V$ is open on level sets of $U$.