A note on Gaussian curvature of harmonic surfaces

TL;DR

This paper investigates the properties of harmonic surfaces with constant Gaussian curvature, demonstrating the non-triviality of their fundamental group and providing examples of topologically distinct harmonic functions sharing the same curvature.

Contribution

It proves the non-triviality of the fundamental group of harmonic polynomial spaces with fixed Gaussian curvature and presents examples of conjugate harmonic functions with identical curvature.

Findings

Fundamental group of harmonic polynomial spaces with fixed curvature is non-trivial.

Existence of topologically nonequivalent harmonic functions with the same Gaussian curvature.

Abstract

It was proved that the fundamental group of the space of harmonic polynomials of degree $n(n \geq 2)$, with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic functions having the same Gaussian curvature.