The regularity problem
The classical Problem of Plateau asks for a surface of least area among all surfaces bounded by a given Jordan curve Γ in three-space. (A Jordan curve is a one-to-one continuous image of the unit circle.) The classical solutions of this problem given in the 1930s by Douglas and Rado showed that there must exist such a surface of least area (without showing how to compute it). The proofs went by picking a sequence of surfaces of smaller and smaller area, and then showing that some subsequence has to converge. There was a hitch though: the limit surface (the one of least area) might acquire branch points. In that case it wouldn't really be a "surface" in the classical sense. This unsatisfactory situation persisted until 1970, when Osserman showed how to modify a surface in the vicinity of an interior branch point to decrease the area. It would be nice to put an animated picture here illustrating this construction. Maybe someday--but Osserman didn't give explicit formulas for his construction, so it's not a trivial task to produce such an animation. The technical jargon for what Osserman proved is the "interior regularity of the solutions of Plateau's problem". (Regularity means, in this context, no branch points, although often the same word is used for smoothness up to the boundary)
Osserman's construction varies the surface in the C0 topology--that means that the modified surface, while close to the original, doesn't necessarily have tangent planes close to the original. This is somewhat unsatisfactory; one would like to be able to vary the surface in such a way as to keep many derivatives close to those of the branched surface. I solved this problem for the C1 topology in
On interior branch points of minimal surfaces, Math Zeitschrift 171 (1980) 133-154.
In that paper, I gave explicit formulas for how to decrease the area by modifying a surface in the C1 topology near an interior branch point. (The paper claims to decrease the area in the Cn topology, and indeed does so in some neighborhood of the branch point, but the smoothing argument in the last section is wrong for n > 1, so the paper does not show how to globally decrease the area in a C1 way). Here it would be nice to see an animation of this construction; I hope to do this someday--it should be easier than for Osserman's construction, since explicit formulas are available.
The question of improving the topology from C0 to Cn is related to the question of whether the regularity applies to only a least-area surface (an absolute minimum of area) or to relative minima of area also. Since the solution of Plateau's problem is not in general unique, and since relative minima of area correspond to physically stable soap films, it is of interest to prove the regularity of relative minima. But in which topology do we define the concept of "relative minimum" of area? If we use the Cn topology, then it is easier to be a relative minimum, the greater n is, since fewer variations are allowed. Consequently, regularity for relative minima is a better theorem, the larger n is. An absolute minimum, however, is of course a relative minimum in any of these topologies. Osserman only claimed his theorem for absolute minima, but it works for relative minima in the C0 topology, but only for relative minima in that topology. There is, however, no a priori reason why a Cn relative minimum should be also a C0 relative minimum.
Osserman's construction only works for true branch points. False branch points were ruled out later, by Gulliver and Alt (independently). Neither Osserman's construction nor mine works for boundary branch points. My construction also assumes the branch point is true, so it still relies on Gulliver and Alt to deal with false branch points. In 1973, Gulliver and Lesley were able to show that boundary branch points can't occur in a least-area surface with a real-analytic boundary. The problem is still open in case the boundary is only Cn.
Recently Tromba has given (in A Theory of Branched Minimal Surfaces) new calculations of the higher variations of Dirichlet's integral, with the aid of which he can give yet a third proof of the interior regularity of relative minima, but this time in the Cn topology. His calculations are global, using Cauchy's integral formula, so a smoothing argument is not needed; my arguments were local, using Fourier series. We both find a variation that does not vanish; if the branch point has order m and index k then the variation that vanishes will depend on some power zL in the formula for the minimal surface, where L is not divisible by m+1. If no such L exists, then the branch point is false.