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This is a classic result in differential geometry and is worth mentioning in these posts on minimal surfaces. Before we can talk about the deformation we need a definition.

**Definition.** A minimal surface described by the Weierstrass-Enneper data or has an *associated family* of minimal surfaces given by, respectively, or

The catenoid has Weierstrass-Enneper representation . Thus, the associated family of surfaces of the catenoid has Weierstrass-Enneper representation , which corresponds to the following standard parametrization.

, for any fixed , where

**A very beautiful result in minimal surface theory.** *The catenoid can be continuously deformed into the helicoid by the transformation given above, where represents the catenoid and represents the helicoid. It should be pointed out that the parametrization above represents a minimal surface for any value of That is, any surface in the associated family of a minimal surface is also minimal.*

The surfaces below, plotted for different values of , represent the associated family of minimal surfaces of the catenoid.

I’ve spent a few posts talking about the theory behind minimal surfaces. So what? Lets actually look at some.

Prior to the 18th century the only known minimal surface was the plane. This changed when Jean Baptiste Meusnier discovered the first non-planar minimal surfaces, the catenoid and the helicoid.

The catenoid may be parametrized as . This is a surface of revolution generated by rotating the catenary about the -axis. It is easily checked that the mean curvature of is zero. Thus, the catenoid is a minimal surface. It can be characterized as the only surface of revolution which is minimal. That is, if a surface of revolution is a minimal surface then is contained in either a plane or a catenoid.

The helicoid may be parametrized as . It is easily checked that the mean curvature of is zero. Thus, the helicoid is a minimal surface. It can be characterized as the only minimal surface, other than the plane, which is also a ruled surface. That is, any ruled minimal surface in is part of a plane or a helicoid.

Placing geometric restrictions on surfaces is a common theme in classification in the study of minimal surfaces. For example, assuming a surface has a parametrization of the form , one can explicitly solve the resulting differential equation to find the minimal surface solution , which locally parametrizes Scherk’s minimal surface (discovered by Heinrich Ferdinand Scherk in 1835). Note that although this surface can be realized locally as a graph, its domain of definition is not the entire plane as it must be represented on patches of the form and –.

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.

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.

I have decided to post all of my notes on minimal surfaces. These notes are essentially a summary of how I spent my summer. Each post will build on the previous and there will be four in total. 1 area functional, 2 harmonic function and isothermal coordinates, 3 Weierstrass-Enneper representation, 4 examples.

**Definition. **Let be a real valued function of two real variables and defined on a domain . The partial differential equation

is known as *Laplace’s equation*. If , , , , , , and are all continuous and if satisfies Laplace’s equation then is *harmonic* on .

An interesting relationship between minimal surfaces and harmonic functions comes about when the surface is parametrized by isothermal coordinates

**Definition.** A parameterization is called *isothermal* if and

**Theorem.** Isothermal coordinates exist on any surface .

*Proof. A Survey of Minimal Surfaces *[Osserman].

When isothermal parameters are used, there is a close relationship between the Laplace operator and mean curvature. We have the following formulas for an orthogonal coordinate system

**Theorem.** If the patch is isothermal then .

*Proof.* Since and , we have

By examining the formula for mean curvature when and , we see that

Therefore, .

**Corollary.** A surface with isothermal coordinates is minimal if, and only if, , , and are harmonic functions.

*Proof.* If is minimal then and, by the previous theorem, On the other hand, suppose that , , and are harmonic functions. Then is harmonic so and, by the previous theorem, . Therefore, since is the unit normal and , then and is minimal.

Earlier this month I gave a lecture at the differential geometry seminar on the relation between minimal surfaces and the area functional. This is a summary of the lecture.

**Definition.** A regular parametrized surface is called *minimal* if its mean curvature is zero everywhere.

We shall try to understand why the word “minimal” is used for such surfaces. One of the original motivations for the development of the theory was the study of soap films formed when dipping closed wires into soapy water. These films tend to form surfaces of least area. In 1760 Joseph Lagrange recognized the connection between surfaces of least area and minimal surfaces and proposed the problem of showing the existence of minimal surfaces with a given boundary. This is now known as Plateau’s problem, named after the Belgian physicist Joseph Plateau for his experiments with soap films.

Formally, Plateau’s problem can be stated as follows. Given a curve , find a minimal surface having as its boundary. As we shall see, least-area surfaces are minimal. Thus, another version of Plateau’s problem is to find a least-area surface having as its boundary.

What is a necessary condition that have least-area among all surfaces with boundary ? The answer may be found in a simplified version of the calculus of variations as follows.

Suppose that is a surface of least-area with boundary . Consider the nearby surfaces which look like slightly deformed versions of , Here, is a function on the domain of which has the effect, when multiplied by a small and added to , of moving points of a small bit and leaving fixed. That is, on where is the boundary of the domain of and . Let us parametrize by where , . Recall that the surface area of is given by

Now, taking the derivative with respect to , which passes inside the integral, we obtain

We assumed that was a minimum so . Therefore, setting in the equation above, we get

Now, let and

Computing , , and applying Green’s theorem we then get

since on . Of course the first integral is zero as well, so we end up with

Since this is true for all such , by the fundamental lemma of the calculus of variations, we must have

But this shows that that mean curvature is identically zero! Therefore, we have shown the following necessary condition for a surface to be area minimizing: if is area minimizing then is minimal.

I particularly like the previous result as it has a very nice application of Green’s theorem. Unfortunately, the previous result is somewhat limiting since it only considers if our surface is parametrized as a graph. By dropping this restriction we can arrive at a slightly more general result.

We define the normal variation of a surface in to be a family of surfaces representing how changes when pulled in a normal direction. Let denote the area of . We show that the mean curvature of vanishes if, and only if, the first derivative of vanishes at .

Let be a regular parametrized surface and choose a bounded domain . Suppose that is differentiable and , where is the union of the domain with its boundary . Let denote a unit vector field such that is perpendicular to** **** **for all ; that is, .

**Definition. **The *normal variation* of , determined by is the map given by for and . For each fixed , the map given by is a parametrized surface.

We denote by , , and the coefficients of the first fundamental form of Then the area of is

**Lemma.** We have where denotes the mean curvature of .

**Theorem.** Let be a regular parametrized surface and let be a bounded domain in . Then is minimal on if, and only if, for all such and all normal variations of .

*Proof.* If is minimal then is identically zero and so for any . Conversely, assume that for any , but that there exists some for which . We choose such that and is identically zero outside a small neighborhood of . But then, for the variation determined by this . This contradiction shows that . Since is arbitrary, is minimal.

It should be pointed out that we have said nothing about the second derivative of at , so that a minimal surface, although a critical point of , may not actually be a minimum. This is a question concerning the stability of minimal surfaces and might possibly be discussed in a later post.