Korean J. Math. Vol. 23 No. 3 (2015) pp.439-446
DOI: https://doi.org/10.11568/kjm.2015.23.3.439

Harmonic mapping related with the minimal surface generated by analytic functions

Main Article Content

Sook Heui Jung


In this paper we consider the meromorphic function $G(z)$ with a pole of order $1$ at $-a$ and analytic function $F(z)$ with a zero $-a$ of order 2 in ${\Bbb D}= \{ z : |z| < 1 \}$, where $-1<a<1.$ From these functions we obtain the regular simply-connected minimal surface $S=\{(u(z),v(z), H(z)) : z \in {\Bbb D} \}$ in $E^3$ and the harmonic function $ f = u + iv $ defined on $\Bbb D$, and then we investigate properties of the minimal surface $S$ and the harmonic function $f$.

Article Details

Supporting Agencies

This work was supported by a research grant from Seoul Women’s Univer- sity(2014).


[1] L. V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966. Google Scholar

[2] M. Chuaqui, P. Duren and B. Osgood, Curvature properties of planar harmonic mappings, Comput. Methods Funct. Theory 4 (2004), 127–142. Google Scholar

[3] M. Chuaqui, P. Duren and B. Osgood, Univalence criteria for lifts of harmonic mappings to minimal surfaces, J. Geom. Analysis 17 (2007), 49–74. Google Scholar

[4] J. G. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. 9 (1984), 3–25. Google Scholar

[5] P. Duren and W. R. Thygerson, Harmonic mappings related to Scherk’s saddle-tower minimal surfaces, Rocky Mountain J. Math. 30 (2000), 555–564. Google Scholar

[6] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692. Google Scholar

[7] R. Osserman, A Survey of Minimal Surfaces, Dover, New York, 1986. Google Scholar