The Minimal Surfaces Over the Slanted Half-Planes, Vertical Strips and Single Slit
In this chapter, we discuss the minimal surfaces over the slanted half-planes, vertical strips, and single slit whose slit lies on the negative real axis. The representation of these minimal surfaces and the corresponding harmonic mappings are obtained explicitly. Finally, we illustrate the harmonic mappings of each of these cases together with their minimal surfaces pictorially with the help of mathematica. The content of this chapter is a shorter version of an article of the author’s report of 2011 and published in arXiv (http://arxiv.org/pdf/1204.2890.pdf) in 2012.
KeywordsHarmonic Mapping Minimal Surface Blaschke Product Vertical Strip Strip Domain
The research of Liulan Li was supported by National Science Foundation (NSF) of China (No. 11201130), Hunan Provincial Natural Science Foundation of China (No. 14JJ1012) and construct program of the key discipline in Hunan province. Saminathan Ponnusamy is currently on leave from the Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India. The research of Matti Vuorinen was supported by the Academy of Finland, Project 2600066611. The authors thank the referee for useful comments.
- 2.Dorff, M.: Harmonic univalent mappings onto asymmetric vertical strips. In: Papamichael, N., Ruscheweyh, St., Saff, E.B. (eds.) Computational Methods and Functional Theory 1997, 171–175. World Scientific Publishing, River Edge (1999)Google Scholar
- 4.Dorff, M., Rolf, J.S.: Soap films, differential geometry and minimal surfaces. In: Dorff, M. (ed.) Explorations in Complex analysis, Classroom Resource Material, pp. 85–159. Mathematical Association of America, Washington, DC (2012)Google Scholar
- 9.Hengartner, W., Schober, G.: Curvature estimates for some minimal surfaces. In: Hersch, J., Huber, A. (eds.) Complex Analysis, pp. 87–100. Birkhauser, Basel (1988)Google Scholar
- 11.Jun, S.H.: Mappings related to minimal surfaces. J. Chung. Math. Soc. 16(4), 313–318 (2006)Google Scholar
- 15.Ruskeepää, H.: Mathematica Navigator, 3rd ed. Academic Press, Boston (2009)Google Scholar