Michael Beeson's Research

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Enneper's Surface

Enneper's surface was discovered in 1864 by Alfred Enneper, who turned 34 that summer. This was eight years after his Ph. D. under the supervision of Dirichlet at Göttingen, where Enneper lived his entire life, from student to Professor Extraordinarius. Enneper's surface is defined in the entire complex plane, so it is an example of a complete minimal surface (no boundary). However, we are interested in considering portions of it, defined in a disk of radius R. Then it is bounded by ``Enneper's wire'', $$\def\vector#1#2#3{\left[ \begin{array}{l} #1\\ #2\\ #3 \end{array} \right]} \Gamma_R(\theta) = \vector { R \cos \theta - \frac 1 3 R^3 \cos 3 \theta} { - R \sin \theta - \frac 1 3 R^3 \sin 3 \theta} { R^2 \cos 2 \theta} $$ The same formula, with r in place of R, defines Enneper's surface in polar coordinates. Click the links below to see pictures for three values of R.

Enneper's surface, with R=1 . This is the critical value of R, at which Enneper's surface has zero second variation in one direction. For R > 1, there are two more minimal surfaces bounded by Enneper's wire.

Enneper's surface, with R=1.2 . You would not be able to see this surface in physical soap film (unless you put in additional supporting wires), because is unstable. That is, some very nearby surfaces have smaller area.

Enneper's surface, with R=1.7 . Here R is almost as large as it can be and still have no self-intersections (of the surface or the boundary).

One question of interest is this: given R, how many minimal surfaces are bounded by Enneper's wire \(\Gamma_R\)? The answer is: for R less than or equal to 1, exactly one. For R between 1 and \(\sqrt 3\), the answer is exactly 3. This result has several different proofs for \(R\) in different ranges.

When \(R < \frac 1 {\sqrt 3} \)

Then the projection of \(\Gamma_R\) on the \(xy\) plane is convex. According to a theorem of Radó, if the boundary curve has a convex projection on a plane, then the surface is a graph $z= f(x,y)$ over that plane; and then \(f\) satisfies a differential equation that has a unique solution for given boundary values. Radó proved this in 1930, using Hopf's 1927 proof of uniqueness for the differential equation.

When \(R < 0.882\)

Then the total curvature of \(\Gamma_R\) is less than \(4\pi\). J.C.C. Nitsche proved in 1973 that any boundary curve (including polygons) with total curvature less than \(4\pi\) is a curve of uniqueness for Plateau's problem. The total curvature of Enneper's wire is given by a complete elliptic integral of the second kind, so it can only be computed numerically. Nitsche's proof uses the eigenvalue problem associated with the second variation of area, which was well understood long before 1930, but Barbosa and do Carmo proved a new result about that eigenvalue problem, published in 1974, and Nitsche was so quick to apply it that his publication date was a year earlier.

When \(R \le 1\)

Ruchert proved in 1981 that uniqueness for Enneper's wire holds up to and including \(R=1\).

When \(R\) is slightly greater than 1

When \(R > 1\), Enneper's surface is not a relative minimum of area, because the second variation of area is negative, as shown by Schwarz (Radó page 39 references Schwarz's collected works, without a date); but it seems Radó applied Schwarz's theory to Enneper's surface in 1930. Therefore (since Plateau's problem is known to be solvable with an absolute minimum of area) \(\Gamma_R\) bounds at least two minimal surfaces. An argument by symmetry shows that there must then be at least two absolute minima. No formula is available to compute these surfaces; Nitsche computed the first few Fourier coefficients of them and studied the bifurcation process by which Enneper's surface splits into three surfaces as \(R\) increases through 1.

Until 1982, it was not known that there are no more than 3 solutions. In that year, Beeson and Tromba introduced another parameter (besides \(R\)) and identified the bifurcation as part of an instance of the cusp catastrophe of Thom. This was the first rigorous identification of the cusp catastrophe in an infinite-dimensional context. It follows that in some neighborhood of Enneper's surface there are exactly three solutions, for \(R\) slightly greater than 1. (We did not give an explicit estimate on how much larger.) It follows from Ruchert's theorem and a simple compactness argument that for \(R\) slightly larger than 1, \(\Gamma_R\) bounds exactly three minimal surfaces. (Although Ruchert's paper is listed in our bibliography and we knew this consequence of his theorem, we failed to point it out explicitly in the paper.)

When \(1 < R < \sqrt 3\)

In 2016, Beeson proved that for all \(R\) between 1 and \(\sqrt 3\), there are exactly three minimal surfaces bounded by \(\Gamma_R\). The paper can be found here.

When \(R \ge \sqrt 3\)

Enneper's wire has self-intersections, and Plateau's problem is usually considered only for Jordan curves. Beeson's proof does not apply when \(R \ge \sqrt 3\).

Calculations to prove the above results (some using the computer algebra system Sage) can be found here.