The famous Prato experiment is to put the wire in the closed curve into the soap solution and then take it out. Due to surface tension, it is covered by a film with the smallest surface area. This surface with the smallest surface area is the so-called minimal surface, and the problem of finding this membrane surface mathematically is called Prato problem. This problem can be solved by variational method.

From the point of view of variational theory, we can consider the normal variation of the surface with the known closed curve γ as the fixed boundary. According to Euler-Lagrange equation (see variational method), for any such variation, the necessary and sufficient condition for the surface area to reach the critical value is the average curvature of the surface, Hⅹ0. Therefore, this geometric condition is usually used to define the minimum surface.

In the three-dimensional Euclidean space E3, if a surface can be expressed by the equation z=z(x, y), it is called a graph or nonparametric surface. According to the minimax condition H = 0, z(x, y) of E3 minimax graph satisfies the following second-order nonlinear elliptic differential equation:

It is usually called the minimum surface equation.

Important examples of E3 minimum surface are: ① the minimum developable surface is a plane; ② The smallest ruled surface of non-plane is a regular helicoid; ③ Catenary is the only smallest surface of revolution; ④ The minimal surface whose curvature line is a plane curve is Ennapel minimal surface; ⑤ Schaekel minimal surface is a minimal helicoid, which can be regarded as a translational minimal surface with real generating curve. Generally speaking, the coordinates of E3 minimal surface can be expressed as a harmonic function of isothermal parameters (parameters that make E=G and F=0 in the first basic form of surface). E3 has no compact minimal surface without boundary.

Historically, the development of minimal surfaces has revolved around the Prato problem, which is essentially a nonlinear elliptic boundary value problem. As early as 1930 ~ 193 1 year, T. Rado and J. Douglas independently solved this problem within the scope of generalized solution, and they obtained the following existence theorem: given any Jordan closed curve, there is always a generalized minimal surface with γ as the boundary. There may be isolated branch points where the surface is not immersed. It was not until 1970 that R. Osman proved that the solutions of Rado and Douglas are internally regular everywhere, that is, there will be no bifurcation points. Later, Qiu Chengtong and others solved the problem of when immersion became embedding.

In addition to this kind of existence problem, there are many unique problems, the most famous of which is Bernstein theorem: a completely minimal graph in E3 must be a plane.

Just as the derivative is used to determine the extreme value of a function, the first-order variation of the area functional is divided into zero, which is only a necessary condition for the minimum area. In order to further determine the surface with the smallest area, the second variation must be considered. Under any normal variation, the minimal surface that makes the second-order variation of the area functional non-negative is called a stable minimal surface. The minimum graph of E3 is stable. Therefore, the conjecture naturally comes from Bernstein theorem, that is, the completely stable minimal surface in E3 is a plane. This proposition has been proved by D. Fischer-Cobb and R. Schon, and later by M. Ducamo and Peng respectively. For the gEneralization of Bernstein's theorem in high-dimensional space, people have long raised the question: Let en be a complete minimal hypersurface, then the function z(x 1, x2, …, xn) must be linear? 1965, E. Di Qiaoji proved that n=3 is right; 1966, F. J. Amgren proved that n=4 is also correct. In 1967, J. Simmons proved that everything is right when n≤7. Unexpectedly, in 1968, E. Bombieri, E. Di Giorgio and E. Giusti jointly proved that when n=8, it is wrong. So, this is a very interesting question. Chen Shengshen, Xiang Wuyi and Qiu Chengtong. They have made important contributions to minimal surfaces and their generalization in high-dimensional manifolds.