Weierstrass Representation of Minimal Surfaces and Related Basic Quantities

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.