Minimal surface
A minimum surface is a surface in space, the local minimum surface area has. Such forms take example of soap films, if they ( such as a blowing ring ) stretched over a corresponding frame.
In mathematical language minimal surfaces are the critical points of the area functionals:
Here are the sizes and is declared (see Hessian matrix ). Note that a minimum area is not necessary minimum surface area, but rather is a stationary point of the surface area functional. One can show that the disappearance of the first variation of the surface area of functionals in two spatial dimensions equivalent to the disappearance of the middle curvature, if the manifold is considered sufficiently regularly.
Minimal surfaces are since the 19th century the focus of mathematical research. A major contribution was the experiments of the Belgian physicist Joseph Plateau.
- 2.1 The minimum area of H. F. Scherk
- 2.2 The catenoid
- 2.3 The helical surface
- 2.4 The Henneberg - surface
- 4.1 Maximum Principle
- 4.2 uniformizing pictures
- 4.3 Analytical Reell character
- 4.4 Complete sets of Amber and Liouville
- 4.5 The area of a minimal surface
- 5.1 Complex representation
- 5.2 Integral representation
- 5.3 Integral representation Free
- 6.1 Derivation and Parameterinvarianz
- 6.2 The one-and two -dimensional content
Remarks on existence theory in two variables
A two-dimensional parameter range is always a special dar. Because with the tools of the theory of functions can be achieved much more detailed statements than in higher space dimensions. This way you can always withdraw, for example on the disc as a parameter region with the Riemann mapping theorem. Also, the uniformization theorem applies only in two spatial dimensions. It allows to introduce isothermal parameters which are required for the solution in the parametric case. That is why the theory in two variables is also developed very far.
Formulation as a variational problem
Now, a surface is a minimal surface if and only if the mean curvature is zero at every point. This raises a minimal surface as a special case of a surface of prescribed mean curvature dar. This also shall not deprive the calculus of variations, they are minima of the Hildebrand Chen functionals
The Euler equations as a necessary Minimalitätsbedingungen this functional are named after Franz Rellich H- surface system
Here, the average curvature.
Parametric case
For this functional, the question arises of the existence of local minima for a given continuous boundary curve of finite length. This task is often called a plateau cal problem in the literature. Assuming a smallness condition on the mean curvature, which is always satisfied in the minimal surface case, this question can be answered positively. To be convinced of this, minimizing at the same time and the energy functional
Introduction of so-called almost -isothermal parameters. In 1884, Herrmann Amandus Schwarz proved the following theorem: In the set of continuously differentiable, simple closed, orientable surfaces of genus zero (ie no holes), the sphere is the area which defines the largest volume for a given surface area.
Branch points
Points at which meets the solution which is known as branch points. Branch points are so interesting, because at these points the parameterization can be singular. Or worse yet is the second possibility, it could also be that the solution is locally no more space, but only one curve. Now provide functional theoretical considerations which are essential inspired by the work of Carleman and Vekua that the solution can at most finitely many such branching points. Unfortunately, the above method does not exclude such branching points a priori. Only with the complex set of Gulliver -Alt- Osserman this is possible a posteriori. That is why there is a desire to solve the problem Plateausche in the class of branch- point -free H- space. That is until today an open question.
Non-parametric case
The above method does, but only for constant success. Depends on the mean curvature of addition of the solution, so you can still do something in the case of a graph. If a graph is, he writes as and the function satisfies the non-parametric equation of prescribed mean curvature
A low-lying Existenzresulat provides the solvability of the Dirichlet problem of this partial differential equation also assuming a smallness condition and other technical requirements. The uniqueness is also clarified by a maximum principle for the difference of two solutions. In addition, graphs are due to
Always branching point free.
Examples of minimal surfaces
Here are several examples of various minimal surfaces are given in three-dimensional Euclidean space. Some of them can not embed without self-intersections in the three-dimensional space you. Others are not on the edge of its domain of definition ever be continued, as the first example shows.
The minimum area of H. F. Scherk
The minimum area of Heinrich Ferdinand Scherk (1835 ): We are looking for all solutions of the non-parametric minimal surface equation which can be written in the form and comply with the conditions. We use this structure first in the minimal surface equation and obtain
Equivalent switching supplies with a
According to the theory of ordinary differential equations exists exactly one solution for the initial value problems
And
These solutions loud
And
It remains to be noted that we could vary the initial values and with a. However, you can Without loss of generality because of the structure condition and the fact that the functions themselves do not occur in the ordinary differential equations require. Thus we obtain
We notice that this minimum area to the squares
Is explained and not beyond resumable. This surface can be embedded as a graph in three dimensions.
The catenoid
If you can rotate the chain line to the x - axis, which also gives an embedded minimal surface in three-dimensional space - a catenoid. Catenoid are the only minimum surfaces that are surfaces of revolution at the same time. They meet at a positive parameter c > 0, the equation
She was one of the first experimentally observed plateau of solutions of Plateau 's problem. Here, the edge data were two circular rings, which form the top and bottom curve of a truncated cone or cylinder.
The catenoid minimal surface as derived from Leonard Euler to 1740.
The helical surface
Closely related to the catenoid, the helicoid or spiral surface. It emerges from a catenoid by a discontinuous but isometric deformation. For a parameter c > 0 it satisfies the equations
And this minimum area is embedded in the three-dimensional space.
The helical surface as a minimal surface is by Jean -Baptiste Meusnier de la Place (1776 ).
The Henneberg - surface
The Henneberg - surface is an example of a minimal surface, which is the image of an immersion into the three-dimensional Euclidean space, but can not be embedded in three-dimensional Euclidean space. Are your defining equations
In addition, this surface is not orientable, speaking clearly, you can not decide which side of this surface above and what is below.
It is named after Lebrecht Henneberg, who introduced it in his dissertation in 1875.
Higher spatial dimensions
In high dimensions of space, access to the Plateau 's problem is more feasible. Here we have only the possibility of conceiving the solution as a graph. The minimal surface equation for the graph enrolls
Through the theory of weak solvability of elliptic boundary value problems can be in this situation to guarantee the existence of solutions. Subsequent Regularitätsbetrachtungen provide a classical solution. As in two spatial dimensions, the uniqueness is obtained here by a maximum principle for the difference of two solutions.
Some interesting statements about minimal surfaces
Due to the relatively simple structure of the equations to be met by minimal surfaces, can be transferred to minimal surfaces in two variables a number of well-known statements which are known particularly for holomorphic or harmonic functions.
Maximum principle
For a minimal surface, the inequality holds
The minimum area takes its maximum ie on the edge of the area on which it is declared to.
Uniformizing pictures
In geodesy, one can introduce so-called isothermal parameters. The figure, which is called the uniformizing accomplished figure. Uniformizing pictures of minimal surfaces are harmonic functions.
Reell analytical character
Minimal surfaces are, as long as they are in isothermal parameters, real analytic functions inside the district in which they are declared. This means that the parametric representation can be developed in each point of the domain in a neighborhood of this point in a convergent power series. Thus, it is infinitely differentiable. Is beyond the boundary curve at a point real analytic, then the minimal surface can be continued in a neighborhood of this point real analytic over the edge.
The sets of amber and Liouville
The set of Sergei Bernstein for minimal surfaces is: An all over the stated solution of non- parametric minimal surface equation necessary satisfies the equation
With constants.
From this theorem follows immediately the Liouville's theorem for minimal surfaces: A limited throughout the stated solution to the nonparametric minimal surface equation satisfies necessary
The area of a minimal surface
The area of a minimal surface with unit normal writes in the form
It must be assumed that the boundary curve simply closed and is continuously differentiable.
Representation formulas
To better understand minimal surfaces it is sufficient to consider not only the differential equations which they satisfy, but you should also identify specific representations of the solution.
Complex representation
Introduction of isothermal parameters u and v, we first obtain the H- surface system for H = 0
Thus the second-order equation writes in the form
With the complex variable and and we obtain the representation
We call a complex curve which satisfies the conditions and an isotropic curve. Next we call a surface which can be written in the form of a sliding surface. A generalized definition of minimal surfaces is the following: A minimal surface is a sliding surface, whose generators are isotropic curves.
Real minimal surfaces then satisfy the conditions
Integral representation
Named after Karl Weierstrass and Alfred Enneper representation formula provides a link between differential geometry and the theory of functions. Now Weierstrass has had great influence on the emergence of the theory of functions. This representation formula was one of the reasons why this relatively new branch of mathematics was taken seriously and was so successful and is. He has found that every non-constant minimal surface can be written as the integral of g and h with the two holomorphic functions. More precisely, for the components
This representation formula makes it possible to generate images of any minimal surfaces with the help of modern computer algebra systems. For example, some images of minimal surfaces in this article using these formulas with the Maple program were created.
Free integral representation
Since we have seen that there are to integrate the differential equation H = 0, isotropic curves is sufficient to determine, we obtain for real minimal surfaces, the so-called integral representation loose
With a holomorphic function which must satisfy the condition. Levels thus evade this view! In order to now the significance of the complex variable w to clarify for a real minimal surface, provides a lengthy account
Here is the unit normal vector of the minimal surface. To summarize: By specifying the complex number w and 1 / w is the unit normal vector of the minimum area determined. Conversely depends w and 1 / w solely upon. The statements of this section are in particular the book by W. Blaschke and K. Leichtweiß Elementary refer differential geometry, see also literature.
Remarks to the area of functional
We will first derive this functional generally and show the invariance under positively oriented parameter transformations. Finally, we will explicitly calculate the one-and two -dimensional special cases.
Derivation and Parameterinvarianz
We note that our minimal surface can interpret in the n- dimensional real vector space as an m- dimensional manifold. This is always possible due to the embedding theorem of Nash. We will first explain the metric tensor
With determinant
We recall that there is a content of an m-dimensional surface as m -dimensional integral of the characteristic function of this area. A characteristic function is everywhere identical one on the lot and otherwise identically zero. Thus we only need to express the surface element suitable. We begin and explain in a fixed point u is the tangent
Further we choose vectors such that the system
Is positively oriented and satisfies the two conditions and for all reasonable values of i and j. Thus, the surface element writes
If we note the following theorem on the determinant
Then we can use our surface element in the form
. Write Using the transformation formula, we now define the invariance under the same direction parameter transformations of the surface element and thus the surface area functional.
The one-and two -dimensional content
In one space dimension, this functional reduces to the ordinary path. It is
Did you present a two-dimensional surface which is embedded in the three-dimensional space, we obtain the identity of Lagrange:. This means that the area functional writes