Michael Beeson's Research on Minimal Surfaces
While I was a graduate student at Stanford (1967-71) I learned something of the beautiful and classical theory of minimal surfaces from Robert Osserman and Robert Finn, as well as indirectly from their students Joel Spruck and David Hoffman. A few years later, while studying constructive mathematics, I made a case study by trying to give a systematic constructive treatment of minimal surfaces. Several of the fundamental theorems present diffficulties from the constructive point of view, for example the fundamental existence theorem that says every Jordan curve in three-space bounds a surface whose area is minimum among surfaces with the given boundary. This is called the solution to Plateau's problem. The usual proofs of this theorem are non-constructive, i.e. they fail to provide an algorithm for computing a solution.
In my logical studies, I was working on the relationship between constructive existence proofs and continuity. I formulated a general principle that if we have a constructive existence proof, the thing constructed must depend continuously on all parameters, at least locally. That is, solutions can be continued locally, but perhaps not globally. This principle, which I verified metamathematically for various systems, can be used in reverse to show that a certain theorem is not constructively provable: if the solution does not depend continuously on parameters, the theorem isn't constructive. For example, the theorem that a uniformly continuous function on [0,1] has a minimum is not constructive, because the minimum can shift from near x = 1/4 to near x = 3/4 suddenly as a parameter passes through a critical value. If you want to compute the minimum of a function, you will need more information than just its modulus of uniform continuity.
I wanted to show that Plateau's problem (the existence of a minimum of area with a given boundary) is essentially non-constructive. Using the method described above, it would suffice to show that the minimum-area solution does not depend continuously on the boundary curve. This is the theorem proved in . The curves in question are approximately the shape of the seam of a tennis ball. When the curve is perfectly symmetrical, there are two different solutions, both of minimum area. When the boundary curve is slightly varied, one or the other of them will have smaller area, so the absolute minimum of area jumps from one to the other discontinuously. Since these surfaces are not given by explicit formulas, some non-trivial mathematics was required to prove these geometrically evident facts rigorously. Note: it is still an open question whether a local minimum of area, or even a stationary point of the area functional, can be constructively proved to exist.
In the fall of 1975, I came to Stanford to spend a year as a visiting assistant professor. This gave me the opportunity to work closely with Solomon Feferman on my logical research, but it also brought me back into contact with Robert Finn and Robert Osserman, and the office I had happened to be between Menahem Schiffer and Stefan Hildebrandt, both of whom were experts on minimal surfaces, and kind enough to talk to me. I began to study the work of Garnier, a Frenchman who had apparently proved some partial results, working towards a solution of Plateau's problem (the main existence theorem mentioned above), in the late twenties. Garnier's method, I hoped, was more constructive than the "final solution" given by Douglas and Rado in 1931. Garnier's method involved approximating the curve by polygons, so I became interested in the behavior of minimal surfaces bounded by polygons. It became apparent the most fundamental properties of such surfaces were as yet unproved. For example, that the normal to the surface extends continuously to the corner and is (at the corner) normal to the plane determined by the two polygonal sides. I formulated and successfully proved an analytic represention theorem for the asymptotic behavior of a minimal surface in a corner. This was published in .
My old friend Tony Tromba had written many papers about his approach to infinite-dimensional Morse theory. Seeking to make the virtues of his work more apparent, he wanted to work out the implications fully in a particular case, and he chose the example of minimal surfaces. I studied his work carefully, and proposed to him a problem suggested in an unpublished manuscript by Ian Stewart: to show that the minimal surfaces obtained from Enneper's surface by a two-parameter family of boundary perturbations could be described by the cusp catastrophe of Thom. At the time catastrophe theory was very much in fashion, but nobody had rigorously identified the cusp catastrophe in an infinite-dimensional setting. Discussing this problem, we saw that it fit nicely into Tony's abstract framework, but reduced to proving that certain complicated Frechet derivatives were positive. I said, "I can do those calculations", and the result was .
In the summer of 1977, I joined my old friend Joel Spruck in Bonn for a summer of work on minimal surfaces, at the invitation of Prof. Stefan Hildebrandt. There, emboldened by my success with computing the second variations of Dirichlet's integral for a particular surface in , I began to try to give a direct proof of the regularity of surfaces which furnish a local minimum of area. This theorem (absence of branch points in a minimum of area) was Prof. Osserman's most famous result, a result which he had proved not by calculation but by a very geometrical cut-and-paste argument. Looking at the matter in Tony's abstract setting, it seemed to me that it ought to be solvable by explicitly giving a "direction" in which the Dirichlet integral would decrease, if not to second order, then to some order. This intuition proved to be correct, although it took me most of the summer to complete the calculations. In the end there was a double sum involving binomial coefficients, which for some time I could not add up. It turned out that the key was to generalize it, replacing two occurrences of m by different variables and proving it by induction on one of them, a trick I learned from the chapter on binomial coefficients in Knuth's Art of Computer Programming. This work was published in . Only the interior branch point case was dealt with. The result is a somewhat stronger theorem than Osserman's, because it applies to relative minima in the C[k] topology; in other words, the surface only has to be a minimum of area among all surfaces nearby in the C[k] sense, not the C sense as in Osserman's theorem, and if the surface is not a relative minimum, the theorem gives a one-parameter family of surfaces nearby in the C[k] sense with smaller area, while Osserman's proof is restricted to C. When one is looking at function spaces with the C[k] topology, as one does in Tromba's approach the subject, this is the result one needs. This was, in fact, why I was looking at the problem to begin with, because Osserman's theorem didn't quite state what was needed in that context.
In the winter 1977-78, I continued to study Tromba's work on minimal surfaces, and began to work on the "finiteness problem", which is to prove (or disprove) the conjecture that no Jordan curve in three-space bounds an infinite number of relative minima of the area functional. (It may even be true that no Jordan curve in three-space bounds an infinite number of minimal surfaces, but that is probably much harder to prove.) At Prof. Hildebrandt's invitiation, I returned to Bonn in May to stay 15 months, where I continued to work on this problem. Placing the problem in the general setting developed by Tony Tromba, R. Boehme, and Fritz Tomi, it was not hard to prove that if the theorem is false, there is a C[k] one-parameter family of minimal surfaces u(t) witht he same boundary, such that for t in some interval [0,a] the surface is a relative minimum of area, and for t=0, the surface has a branch point (and hence is not a relative minimum of area, by the result discussed in the previous paragraph). My idea was to analyze the Gauss map of this one-parameter family of surfaces and show that the area of its image (which must be large at t=0 because of the branch point) must also be large for small positive t, which is not possible for a relative minimum of area, because there is a well-known eigenvalue problem associated with the second variation of area, and relative minima correspond to the least eigenvalue, so the image of the Gauss map must not have too large an area. I carried this program through for the case of an interior branch point at t = 0 in the fall, but there was still one remaining obstacle. Tony Tromba's analysis of the Dirichlet functional exposed the existence of the "forced Jacobi fields"; these are "directions" (variations) of a branched minimal surface in which the Dirichlet functional must have zero second variation. They are similar to the three "conformal group" directions, but they apply only to a branched surface. An analysis of these fields had to supplement the main argument. Desperate at the icy winds of the Bonn winter, in February I took the train to Switzerland for a few days, and sitting in a warm cafe there I completed the proof in the interior-branch-point case. This work was eventually published in . The work just described shows that in any case in which we can somehow rule out boundary branch points, we have a finiteness theorem for relative minima of area. One such case is when the boundary curve lies on the boundary of some convex body. This condition is stated as the main theorem of , since it is nicer to state a theorem with purely geometric conditions, rather than the ugly hypothesis that no relative minimum of area has a boundary branch point.
Living alone in a comfortable apartment near the Rhine in Bad Godesberg, I had time for a sustained attack on the boundary branch point case. As the German winter of 1978-79 dragged on and on, each day I covered pages with fresh calculations. Consider then a boundary curve which hypothetically bounds an infinite number of relative minima of area. Then there is a one-parameter family of such relative minima (for t>0) such that the limit surface u(0) has a boundary branch point. The main result of  is that the variation direction du/dt cannot be a forced Jacobi direction. The method of proof is to analyze the behavior of the surface u(t) in the vicinity of the branch point at z=0. For small positive t, the zeroes of the functions in the Weierstrass representation (which are at z=0 when t=0) migrate away from the origin. Extensive calculations reveal how the eigenfunction of the the area functional depend on these zeroes. Their positions determine the Gaussian image of the surface for small positive t. That Gaussian image will have "extras spheres" or "extra hemispheres" to provide the extra area required by the Gauss-Bonnet formula. It must be hemispheres only, or the argument of  will apply. The idea in  is to calculate directly the dependence of the least eigenfunction on the zeroes, to show that it cannot be of one sign, as the least eigenfunction must be. Eventually this dependence was reduced to a set of polynomial equations. If those equations have no solution, we're done. I reached this point sometime in the spring of 1979, and used a hand calculator to attack special cases (low order and index of the boundary branch point). The simplest cases turned out to have no solution, and I was encouraged, but then I discovered an order and index for which the equations did have a solution. I can remember seeing these results in the red LED of the calculator as they were checked and double-checked.
This outcome was not what I had wished for. It means that a purely local analysis of the minimal surface cannot succed. In other words, there does exist a one-parameter family of minimal surfaces, bounded by a straight line instead of a Jordan curve, which locally looks like a counterexample. If indeed such a family cannot be bounded by a Jordan curve, some additional idea will be required to prove it. Realizing this, I decided to write up the calculations so far. The results just discussed were published in. Since the equations I had derived do not have solutions for low order and index of the hypothetical boundary branch point, if the geometry of the curve is such as to rule out such branch points, we do get a finiteness theorem with a purely geometrical hypothesis. Such a condition, for example, can be stated as follows: every point on the boundary curve which is not on the convex hull of the boundary curve has the property that every plane through it meets the boundary curve in at most 8 points. Note that this generalizes the condition of , that the boundary curve lie on the boundary of a convex body.
Looking for purely geometrical hypotheses that would permit drawing a finiteness theorem from , I noticed that the case when the (real-analytic) boundary curve has total curvature at most 6 pi could be treated. J.C.C. Nitsche had already proved that such curves cannot bound an infinite number of minimal surfaces, under the additional hypothesis that the boundary curve does not bound any minimal surface with a branch point. Note that Nitsche's theorem refers to all minimal surfaces, not just relative minima of area. Because of this difference, removing the branch-point hypothesis from theorem is not just a corollary of . There was an additional calculation to be done: I had to prove the impossibility of a one-parameter family of minimal surfaces u(t) all of whom have branch points and such that du/dt is a forced Jacobi direction for every t. This calculation is the main piece of mathematics in .
In the fall of 1979, I moved to Utrecht, to work on logic in the research group of Prof. van Dalen. This marked the end of my research in minimal surfaces for twenty-one years. In calendar year 2000, I had leave from my University work, and found the time to return to the branch point calculations. I asked myself, what distinguishes a straight line from an analytic boundary curve? The answer is that some n-th derivative of the curve is nonzero. Indeed, if the curve does not lie in a plane, each component has some nonzero higher derivative. Using these integers, I was able to derive geometric bounds on the index of the branch point. Armed with these estimates, I was able to push the branch point calculations through. Along the way I discovered that there was a (non-trivial but correctable) error in the 1982 calculations, as well. In between my teaching and my research on automated deduction, it took until August, 2005 to complete all the calculations and triple-check them. The end result is about fifty pages of calculations proving the theorem I had tried to prove twenty-five years ago: a real-analytic Jordan curve in three-space cannot bound infinitely many relative minima of area [of the topological type of the disk]. This work is now on the web in .
Current interests in minimal surfaces.
There are many possible ways to extend the finiteness result. To mention just the more obvious ones: what about topological types other than the disk? I conjecture that it works for orientable surfaces of finite genus. Brian Smith conjectures that the finiteness property fails for non-orientable surfaces in general, and draws a picture of a possible counterexample; but it still might be true for the Moebius band. The second obvious question is, what about polygonal boundaries? Although many papers about minima minimal surfaces with polygonal boundaries have been written, there are still fundamental facts unknown--for example, we still don't have a good Morse theory of minimal surfaces with polygonal boundaries, despite various efforts in that direction over the past twenty years. I am not pursuing these questions now, because my current research work is in automated deduction, but I hope to return someday.