Enneper-Weierstrasse construction

The Weierstrasse representation , sometimes also Enneper-Weierstrasse or Weierstrasse-Enneper construction , is a representation of minimal areas named after Karl Weierstrasse or Alfred Enneper . The latter are regular surfaces in real vector space , which occur naturally as soap-skin surfaces , and therefore "real" structures. It may therefore be surprising that holomorphic functions come to light when they are described, as is the case with the representation to be discussed here. ${\displaystyle \mathbb {R} ^{3}}$

Enneper-Weierstrasse depiction

Let it be a simply connected set ${\displaystyle U\subset \mathbb {C} }$

${\displaystyle z_{0}\in U}$,   , ${\displaystyle c_{1},c_{2},c_{3}\in \mathbb {R} }$

${\displaystyle F:U\rightarrow \mathbb {C} }$ a holomorphic function different from the null function

${\displaystyle G:U\rightarrow {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$a meromorphic function ,

so that the product is holomorphic, that is, has a definable gap in the definition at all poles of . Set ${\displaystyle FG^{2}}$ ${\displaystyle G}$

${\displaystyle \varphi _{1}(\zeta ):={\frac {1}{2}}F(\zeta )(1-G^{2}(\zeta ))}$,

${\displaystyle \varphi _{2}(\zeta ):={\frac {\mathrm {i} }{2}}F(\zeta )(1+G^{2}(\zeta ))}$,

${\displaystyle \varphi _{3}(\zeta ):=F(\zeta )G(\zeta )}$,

Then it's through

${\displaystyle f_{j}(x,y)=c_{j}+\operatorname {Re} \int _{z_{0}}^{x+\mathrm {i} y}\varphi _{j}(\zeta )\mathrm {d} \zeta \quad \quad j=1,2,3}$

a parameterization

${\displaystyle f=(f_{1},f_{2},f_{3}):U\rightarrow \mathbb {R} ^{3}}$ given a minimal area.

Conversely, each minimal area can be parameterized locally in this way, that is, data can be found as above, so that those defined thereby parameterize the presented minimal area in a neighborhood of . ${\displaystyle U,z_{0},c_{1},c_{2},c_{3},F,G}$ ${\displaystyle f_{1},f_{2},f_{3}}$ ${\displaystyle (c_{1},c_{2},c_{3})}$

Here is the real component production , the integral of after is along any path of integration in form to, because of the presumed simple relationship, the value does not depend on the chosen path of integration of the integral. ${\displaystyle \operatorname {Re} }$ ${\displaystyle z_{0}}$ ${\displaystyle x+\mathrm {i} y}$ ${\displaystyle U}$

additions

The above illustration comes from Karl Weierstrass from 1866, around the same time, equivalent formulas were used by Alfred Enneper and Bernhard Riemann .

In the above theorem, the converse yields the existence of a certain parameterization of a minimal area. Often, however, areas are already given in the form of a parameterization, so that the question arises whether the functions and also for a given parameterization of a minimum area can be found. This is generally not the case, but it is if the specified parameterization conforms, that is, if the first fundamental form is a multiple of the identity matrix, more precisely if for a scalar function , where denotes the metric tensor . This becomes clear in the evidence sketch below. ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle (g_{ij})=\lambda \cdot {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$ ${\displaystyle \lambda }$ ${\displaystyle (g_{ij})}$

The pair is called a Weierstrasse representation of the minimal area. In doing so, one often ignores the constants , that is, one mentally shifts the surface so that the zero point lies within the surface. The holomporhen functions meet ${\displaystyle (F,G)}$ ${\displaystyle c_{1},c_{2},c_{3}\in \mathbb {R} }$ ${\displaystyle \varphi _{j}}$

${\displaystyle \varphi _{1}^{2}+\varphi _{2}^{2}+\varphi _{3}^{2}={\frac {1}{4}}F^{2}(1-G^{2})^{2}-{\frac {1}{4}}F^{2}(1+G^{2})^{2}+F^{2}G^{2}=0}$.

If you have reversed three non identically zero, holomorphic functions with given so you can get a holomorphic function and a meromorphic function as found in the sentence, easily one considers that ${\displaystyle \varphi _{1},\varphi _{2},\varphi _{3}:U\rightarrow \mathbb {C} }$ ${\displaystyle \varphi _{1}^{2}+\varphi _{2}^{2}+\varphi _{3}^{2}=0}$ ${\displaystyle F}$ ${\displaystyle G}$

${\displaystyle F:=\varphi _{1}-\mathrm {i} \varphi _{2}}$   and   ${\displaystyle G:={\frac {\varphi _{3}}{\varphi _{1}-\mathrm {i} \varphi _{2}}}}$

do what is requested.

If is constant, then and are obviously proportional and one gets the parameterization of a plane. Many authors rule out this trivial case, and that's what we want to do here. ${\displaystyle G}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{3}}$

example

According to the Weierstrass representation, it is possible to construct minimal areas for given functions and which meet the stated conditions. A very simple and well-known case is the Enneper surface , which is obtained from (constant function) and from . The functions result from the above formulas ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle F(\zeta )=2}$ ${\displaystyle G(\zeta )=\zeta }$ ${\displaystyle U=\mathbb {C} }$ ${\displaystyle \varphi _{j}}$

${\displaystyle \varphi _{1}(\zeta )=1-\zeta ^{2}}$

${\displaystyle \varphi _{2}(\zeta )=\mathrm {i} (1+\zeta ^{2})}$

${\displaystyle \varphi _{3}(\zeta )=2\zeta }$.

So all of them are polynomials whose integration is trivial. As if we choose the zero point, we also set the constants to 0. Then you get for ${\displaystyle z_{0}}$ ${\displaystyle c_{i}}$ ${\displaystyle z=x+\mathrm {i} y}$

${\displaystyle {\begin{aligned}f_{1}(z)&=\operatorname {Re} \int _{0}^{z}(1-\zeta ^{2})\mathrm {d} \zeta =\operatorname {Re} (\zeta -{\frac {1}{3}}\zeta ^{3})|_{0}^{z}\\&=\operatorname {Re} (z-{\frac {1}{3}}z^{3})=\operatorname {Re} (x+\mathrm {i} y-{\frac {1}{3}}(x^{3}+3\mathrm {i} x^{2}y+3\mathrm {i} ^{2}xy^{2}+\mathrm {i} ^{3}y^{3}))=x-{\frac {1}{3}}x^{3}+xy^{2}\end{aligned}}}$

and by similar simple calculations

${\displaystyle f_{2}(z)=-y+{\frac {1}{3}}y^{3}-x^{2}y}$

${\displaystyle f_{3}(z)=x^{2}-y^{2}}$

Therefore is through

${\displaystyle f(x,y):=(x-{\frac {1}{3}}x^{3}+xy^{2},-y+{\frac {1}{3}}y^{3}-x^{2}y,x^{2}-y^{2})}$

Given the parameterization of a minimal area, this is called the Enneper area after its discoverer.

Evidence sketch

The following evidence sketch contains little of the technical details required. The simpler direction starts with the functions and and constructs the conforming parameterization given in the sentence . This procedure was also illustrated using the example of the Enneper area. By utilizing the analyticity, one finally shows that the mean curvature of the surface thus defined disappears and therefore a minimal surface is present. ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle f_{1},f_{2},f_{3}}$

Conversely, if a minimum area is given in parameterized form, the Enneper-Weierstrass representation is determined in the following steps, which essentially represent a reversal of the above construction, whereby an additional difficulty is that one must first obtain a conforming parameterization.

Curvature line parameters

First, the so-called curvature line parameters are determined. This is a parameterization such that the first and second fundamental forms are diagonal in shape. For a patch of area without umbilical points , this is always possible locally by solving a partial differential equation . It then applies , where the normal field and which are the two main curvatures. Since the mean curvature disappears with a minimal surface , it must be. ${\displaystyle f:U\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle \textstyle {\frac {\partial \nu }{\partial u_{i}}}=-\kappa _{i}{\frac {\partial f}{\partial u_{i}}}}$ ${\displaystyle \nu }$ ${\displaystyle \kappa _{i}}$ ${\displaystyle \textstyle {\frac {1}{2}}(\kappa _{1}+\kappa _{2})}$ ${\displaystyle \kappa :=\kappa _{1}=-\kappa _{2}}$

Compliant parameters

In the second step, we construct conforming parameters, see above. We give ourselves a point and move on to a rectangle environment contained in . You can do that because it is a local problem. If again denotes the metric from the first fundamental form, then one considers that the functions , which depend on pairs , actually only depend on one of the variables, by showing that the derivative vanishes according to the other variable. There are therefore real functions with and . The functions are positive and you can use them to define the following figure: ${\displaystyle (x_{0},y_{0})\in U}$ ${\displaystyle U}$ ${\displaystyle U_{0}}$ ${\displaystyle (g_{ij})}$ ${\displaystyle \kappa \cdot g_{ii}:U\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle (x,y)\in U_{0}}$ ${\displaystyle \Phi _{1},\Phi _{2}}$ ${\displaystyle \Phi _{1}(x)=\kappa (x,y)\cdot g_{11}(x,y)}$ ${\displaystyle \Phi _{2}(y)=\kappa (x,y)\cdot g_{22}(x,y)}$ ${\displaystyle \Phi _{j}}$

${\displaystyle \Phi :U_{0}\rightarrow \mathbb {R} ^{2},\quad \Phi (x,y):={\begin{pmatrix}v_{1}(x)\\v_{2}(y)\end{pmatrix}}}$, in which

${\displaystyle v_{1}(x):=\int _{x_{1}}^{x}{\sqrt {\Phi _{1}(x)}}\mathrm {d} x,\quad v_{2}(y):=\int _{y_{1}}^{y}{\sqrt {\Phi _{2}(y)}}\mathrm {d} y}$.

Then a diffeomorphism of the picture and it shows that the three functions ${\displaystyle \Phi }$ ${\displaystyle U_{0}}$ ${\displaystyle V:=\Phi (U_{0})}$

${\displaystyle f_{j}\circ \Phi ^{-1}:V\rightarrow \mathbb {R} ,\quad j=1,2,3}$

form a compliant parameterization of the submitted area.

Holomorphic functions

At this point in the construction there is a conforming parameterization and for the sake of simplicity can be assumed as an open rectangle in the plane. If the plane is identified as usual with the plane of complex numbers, we get three complex functions through ${\displaystyle f_{j}:U\rightarrow \mathbb {R} }$ ${\displaystyle U}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \varphi :U\rightarrow \mathbb {C} }$

${\displaystyle \varphi _{1}(x+\mathrm {i} y):={\frac {\partial f_{1}}{\partial x}}(x,y)-\mathrm {i} \cdot {\frac {\partial f_{1}}{\partial y}}(x,y)}$

${\displaystyle \varphi _{2}(x+\mathrm {i} y):={\frac {\partial f_{2}}{\partial x}}(x,y)-\mathrm {i} \cdot {\frac {\partial f_{2}}{\partial y}}(x,y)}$

${\displaystyle \varphi _{3}(x+\mathrm {i} y):={\frac {\partial f_{3}}{\partial x}}(x,y)-\mathrm {i} \cdot {\frac {\partial f_{3}}{\partial y}}(x,y)}$

The conformity of the parameterization is equivalent to and the minimum area property in this situation is equivalent to the holomorphism of . With the formulas already mentioned above ${\displaystyle \varphi _{1}^{2}+\varphi _{2}^{2}+\varphi _{3}^{2}=0}$ ${\displaystyle \varphi _{j}}$

${\displaystyle F:=\varphi _{1}-\mathrm {i} \varphi _{2}}$   and   ${\displaystyle G:={\frac {\varphi _{3}}{\varphi _{1}-\mathrm {i} \varphi _{2}}}}$

the desired Weierstrass representation is obtained.

Individual evidence

1. ↑ Wolfgang Kühnel : Differentialgeometrie , ISBN 978-3-8348-0411-2 , consequence 3.36: Weierstraß representation
2. ↑ Jost-Hinrich Eschenburg, Jürgen Jost: Differentialgeometrie und Minimalflächen , Springer-Verlag 2014, ISBN 978-3-642-38521-6 , chapter 8.5: The Weierstraß representation
3. ↑ Johannes Nitsche : Lectures on Minimal Areas , Springer-Verlag (1975), ISBN 978-3-642-65620-0 , Chapter III.2.3: The Weierstraß-Enneperschen representation formulas
4. ↑ Johannes Nitsche: Lectures on Minimal Areas , Springer-Verlag (1975), ISBN 978-3-642-65620-0 , historical comment on page 143
5. ^ Wilhelm Blaschke , Kurt Leichtweiß : Elementare Differentialgeometrie , Springer-Verlag 1973, ISBN 0-3870-5889-3 , §46: Curvature lines
6. ↑ Wolfgang Kühnel: Differentialgeometrie , ISBN 978-3-8348-0411-2 , Lemma 3.33
7. ↑ Wolfgang Kühnel: Differentialgeometrie , ISBN 978-3-8348-0411-2 , consequence 3.31
8. ↑ Wolfgang Kühnel: Differentialgeometrie , ISBN 978-3-8348-0411-2 , Lemma 3.35