# Michael Beeson's Research

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## Minimal Surfaces

Minimal surfaces are mathematical surfaces similar to those formed in nature by soap films (not soap bubbles, which have an inside and an outside, with higher pressure inside, but soap films, which have the same air pressure on both sides and are in some equilibrium position because they adhere to some kind of boundary or to each other). This page is not an introduction to minimal surfaces, but rather a page about my research in the subject.

If you have an undergraduate education in mathematics, including two-variable calculus and complex analysis, then you are prepared to learn the mathematics of minimal surfaces. There is a more or less self-contained document Minimal Surfaces that I wrote; but it is not exactly an introduction to the subject, as it does not begin with pictures and examples as a good introduction should. Instead, it is more of a repository of basic results and calculations that you can study in detail once your appetite for the subject is whetted. In other words, it contains the statements, and many of the proofs, of the basic theorems of the subject. It contains, for example, everything you would need to know in order to read my papers.

There are of course many books written by experts of the past and experts of the present, but these books tend to be somewhat encyclopedic and also to take bigger steps in the calculations; the linked exposition is meant to be readable by students.

One particularly interesting minimal surface is Enneper's surface.

Much of my work on minimal surfaces has involved branched minimal surfaces, i.e., minimal surfaces with branch points. These are remarkably simple and beautiful mathematical objects.

Here is an introduction to branch points, with a definition, explanation, and links to animated pictures. If you don't have time for that, here's just one picture.

An important problem in minimal surface theory is Plateau's Problem. In this problem, we are given a Jordan curve C in three-dimensional space, and asked to find (one or more, or all) minimal surfaces bounded by C. We note the following properties that a solution u of Plateau's problem might or might not have:

• u might be an absolute minimum of area among surfaces bounded by C.
• u might be a relative minimum of area among surfaces bounded by C. In that case, it might or might not be an absolute minimum.
• u might be unstable: some arbitrarily nearby surfaces bounded by C have a strictly smaller area.
• u might be isolated: in some neighborhood of u in some suitable space of surfaces bounded by C, there is no other minimal surface.
• u might be stable: the second variation of area D2A(u) is positive in all directions.
• u might have the topological type of the disk, or a more complicated topological type, with or without handles.
• u might be orientable (have a globally defined unit normal), or it might not (e.g., it might have the topological type of the Möbius strip)
• u might be immersed (that is, have no branch points, either in the interior or on the boundary). Sometimes this property is called regularity, but that is confusing, as that term is more often used to mean smoothness up to the boundary.
• u might be immersed in the interior, without saying whether it does or does not have boundary branch points.
• u might be embedded (that is, have no self-intersections)
• u might be non-parametric (that is, can be written in the form z = f(x,y) for some choice of the xy plane)
• u might be regular at the boundary (that is, if the boundary is "smooth" then the surface is equally "smooth" in the closed parameter domain).
• u might be regular in corners (where two straight or C1 pieces of the boundary meet at an angle, there is a nice representation for the surface, implying in particular that the normal extends continuously to the corner)

There are many interesting theorems about each of these properties. Evidently, with 12 properties listed, there are 132 questions about whether one of these properties implies another. Some are trivial, some are easy, some are difficult, and some are still open. Some of the properties we have listed are not quite precise: for example, we haven't specified the metric or topology to be used in defining "relative minimum of area", and we haven't discussed the choices of metrics to be used for "smooth".

Aside from the properties of an individual solution of Plateau's problem, we can ask about the number of solutions for a given Jordan curve C, or about properties that must be possessed by all solutions for a given Jordan curve C. For example, if the total curvature of C is less than 4π, then there is only one solution of Plateau's problem. For a second example, if the boundary curve C has a convex projection on a certain plane, then every solution of Plateau's problem is non-parametric with respect to that plane.

I have personally worked on some of these problems. Of course, you can download my papers with technical exposition and proofs. But I intend to use these web pages to explain the background of the problems, sometimes using pictures, to people with some mathematical background who are not experts. Thus, if you want proofs, see the papers, but if you want pictures and explanations, follow the links.

Can relative minima of area have (interior or boundary) branch points?

Can there ever be infinitely many solutions of Plateau's problem for one fixed Jordan curve C? [Link not yet live, but planned.]

What can or must the behavior of a minimal surface be near a place where the boundary forms an angle, e.g. if the boundary is a polygon in 3-space? [Link not yet live, but planned.]

Finally, one can ask for the structure of the space of all minimal surfaces, as the boundary varies. Tromba and others have obtained deep results in this area, which still haven't been exploited as much as they can be, in my opinion. Tromba and I wrote one joint paper applying these structure theorems to the example of Enneper's surface, and relating the structure of the space of nearby minimal surfaces to Thom's cusp catastrophe. I hope in the future to provide explanations and pictures to go with this work.