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.

## 7 comments

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October 28, 2010 at 7:36 am

WadesworthNice tex man!

October 28, 2010 at 2:29 pm

AllanI went to the comments specifically to complement your LaTeX usage. But it appears that Wade beat me to it.

October 28, 2010 at 5:46 pm

ChiragLOL all those formulas made me dizzy Richard…GOOD JOB. lol.

October 28, 2010 at 7:35 pm

WadeFun Lem FTW btw. (now I actually read your article and didnt just stare at the pretty LaTeX)

October 29, 2010 at 8:46 am

RichardFund lemma and Green’s theorem. Very beautiful

October 28, 2010 at 9:01 pm

BernieYes, yes, of course. Naturally. I approve of everything you wrote.

(lol good job Ricardo)

October 29, 2010 at 12:16 am

thomasthat’s so cool!