Weierstrass Representation of Minimal Surfaces and Related Basic Quantities


  • Weiyi Zhang




Minimal surface; conformal parameterized surface; Weierstrass representation; the first and second basic forms of minimal surfaces.


Minimal surface, as an important surface in differential geometry, has long been one of the research topics of many scholars. It provides far-reaching research materials for geometric analysis and nonlinear partial differential equations, and plays an important role in mathematical general relativity. The minimal surface mentioned in this paper refers to the surface with the smallest area when the boundary conditions remain unchanged, i.e., the Plateau problem. The physical counterpart is the soap film experiment. It is different from a surface of constant mean curvature in another sense. Weierstrass discovered that the general solution of minimal surface equations can be given by complex analysis, that is, Weierstrass representation of minimal surface, thus revealing the essential relationship between minimal surface and holomorphic function and meromorphic function. In this paper, the Weierstrass representation of minimal surface is organized, and the first and second basic forms of minimal surface are derived by using its complex vector form. The study of minimal curved surface has played an important role in many fields such as construction engineering, material science and so on.


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Weihuan C. Minimal Surface. Hunan Education Publishing House, 1993.

Yongxia H. Research on Related Problems in Minimal Surface Modeling. Dalian University of Technology, 2013.

Qunli Z, Pinghuai Z, Xuelin Y, Yi S. Weierstrass Representation and Architectural Modeling of Minimal Surface. Civil and Architectural Engineering Information Technology, 2014, 6(03): 25-38.

Davor D, Željka Š M. Weierstrass representation for lightlike surfaces in Lorentz-Minkowski 3-space. Journal of Geometry and Physics, 2021, 166.

Davor D. Weierstrass representation for timelike surfaces in Minkowski 4-space. Journal of Geometry, 2021, 113(1).

Qingchun J. A Concise Course on Differential Geometry. Science Press, 2021.

Jiagui P, Qing C. Differential Geometry. Beijing, Higher Education Press, 2006.

Yibing B. Preliminary Global Differential Geometry. Hangzhou, Zhejiang University Press, 2006.

Xavier F, Xiaoli C. Modern Minimal Surface Lecture. Higher Education Press, 2011.

Xiangming M, Jingzhi H. Differential Geometry (4th Ed.). Higher Education Press, 2008.

KOISO M. On the stability of minimal surfaces in R3. Journal of the Mathematical Society of Japan, 1984, 36(3).

Kenmotsu K. Weierstrass formula for surfaces of prescribed mean curvature. Mathematische Annalen, 1979, 245(2).

Zidao X, Cunjin S. Weierstrass representations of minimal surfaces in R3 and their first and second basic forms. Journal of Soochow University (Natural Science Edition), 1988(01): 26-30.




How to Cite

Zhang, W. (2024). Weierstrass Representation of Minimal Surfaces and Related Basic Quantities. Highlights in Science, Engineering and Technology, 88, 972-980. https://doi.org/10.54097/4c1d6z95