Harmonic Functions and Parallel Mean Curvature Surfaces
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.
KeywordsHarmonic Function Riemann Surface Curvature Surface Fundamental Form Curvature Vector
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