# Harmonic Functions and Parallel Mean Curvature Surfaces

• Katsuei Kenmotsu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

## Abstract

Minimal surfaces in a Euclidean three space are closely related to complex function theory. A constant mean curvature surface is constructed by a harmonic mapping using the generalized Weierstrass representation formula. In this paper, I present a surface that is constructed by a harmonic function. It is immersed in a complex two-dimensional complex space form with parallel mean curvature vector. We prove that the Kaehler angle function of the surface is obtained by a functional transformation of a harmonic function. And, then, the 1st and 2nd fundamental forms of the surface are explicitly expressed by the Kaehler angle function. As a byproduct, we show that any Riemann surface can be locally embedded in the complex projective plane and also in the complex hyperbolic plane as a parallel mean curvature surface.

## Keywords

Harmonic Function Riemann Surface Curvature Surface Fundamental Form Curvature Vector
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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