Let be a regular parametrized surface and let denote the corresponding complex coordinate. Since and , we may write

We define the complex function as follows,

**Theorem.** Let be a surface with patch and let . Then is isothermal if, and only if, .

*Proof. *Suppose that is isothermal. Note that . Therefore,

Conversely, suppose that . Then and by properties of complex numbers this equations only holds if and which shows that is isothermal.

**Theorem.** Suppose that is a surface with patch . Let and suppose that (i.e. is isothermal). Then is minimal if, and only if, each is analytic.

*Proof.* Suppose that is minimal, then is harmonic; that is, . Thus, . Therefore, each is analytic. Conversely, the same calculation shows that if each is analytic, then each is harmonic, therefore, is minimal.

**Corollary. **.

*Proof.* Since , we may write . Then

We then have that and we can now integrate to get .

The problem of constructing minimal surfaces reduces to finding a tripple of analytic functions with . A nice we of constructing such a is to take an analytic function and a meromorphic function with analytic. Now, if we let and then we have,

Note that is analytic, is meromorphic, and is analytic since . Furthermore, it is easily verified that . Therefore, determines a minimal surface.

**Theorem.** (Weierstrass-Enneper Representation I) If is analytic on a domain is meromorphic on , and is analytic on then a minimal surface is defined by , where

Suppose that is analytic and has an inverse in a domain which is analytic as well. Then we can consider as a new complex variable with . Define and obtain . Therefore, if we replace by and by we get the following.

**Theorem. **(Weierstrass-Enneper Representation II) For any analytic function , a minimal surface is defined by , where

Note the corresponding

This representaion tells us that any analytic function defines a minimal surface.

We can now use the Weierstrass-Enneper representation to produce minimal surfaces. For example, if then we obtain a parametrization for Enneper’s surface. In fact, if then an nth order Enneper’s surface is obtained.

First order Enneper surface

The Weierstrass-Enneper representation leads to infinite families of minimal surfaces and has proved fundamental in relating the study of minimal surfaces to the theory of complex analysis.

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