Not Just a Special Surface

If I had to sum up the content of Hermann Amandus Schwarz’ price winning monograph Bestimmung einer speciellen Minimalfläche from 1867, I would do so using figures from plate VI from the Nachtrag, conveniently compiled in his Collected Works in a single figure:

What is shown here are polyhedra whose vertices are the branched values of the Gauß map of five families of triply periodic minimal surfaces that Schwarz is investigating. 

Schwarz spends most of the over 100 pages discussing a single surface, now called the Diamond or D-surface. It solves the Plateau problem for four consecutive edges of a regular tetrahedron. The details Schwarz provides are overwhelming, and it is easy to overlook that the methods Schwarz develops reach far beyond this special surface, and that he was fully aware of it.

What was keeping mathematicians busy these days? Bernhard Riemann had died in 1866 and left a legacy of new concepts and open problems. Complex analysts and geometers were working towards proofs of the Riemann mapping theorem, the uniformization theorem, and the Plateau problem. Schwarz had its own approach: Solve simple cases first, understand them as well as possible, and then apply the developed methods to solve the general case. Both for the Riemann mapping theorem and the Plateau problem, Schwarz looks at polygonal boundaries. He develops the Schwarz-Christoffel formula, and tries something similar for minimal surfaces.

Schwarz uses cutting edge technology: The Weierstraß representation for minimal surfaces, the language of Riemann surfaces, and elliptic integrals. He realizes that he can do more than just solve Plateau problems: In addition to straight lines, he can also prescribe symmetry planes. This leads to a differential equation which he can solve if the branched values of the Gauß map are sufficiently symmetric.

Competition was fierce, in particular between Göttingen (Riemann and Enneper) and Berlin (Weierstraß and Schwarz). Riemann had left a few pages of notes that hint at what Schwarz discovers. Schwarz must have been shocked when he saw the posthumous paper, with details added by Hattendorf. He also learns that Enneper had used a version of the Weierstraß representation in 1864, maybe without quite grasping its scope, two years before Weierstraß’ note from 1866. It appears that Riemann knew about this, too, as usual. How much did Enneper and Riemann talk in Göttingen? 

With the exception of Schwarz’ figure 47, representing the H-surface, all vertices are antipodally symmetric. I suspect that Schwarz would have instantly nodded if somebody had told him that his differential equation can be solved just under this symmetry assumption, an observation made by Bill Meeks in his 1975 thesis. How the differently symmetric H-surface fits into the picture, together with other, more recently found surfaces like Alan Schoen’s Gyroid, is one of the big open problems of the area.

Scherk’s Fourth Surface

In his second paper about minimal surfaces from 1835, Heinrich Ferdinand Scherk summarizes his earlier findings from 1830 and gives equations for five new minimal surfaces, the first new ones since the catenoid and helicoid.

Equation 7 describes the doubly periodic Scherk surface in general form (the orthogonal case is equation 6). This is the first non-trivial deformation family of minimal surfaces.

doublyscherkshear

Equation 9 is easily recognized as the associate family deformation of catenoid to helicoid, parametrized as screw motion invariant surfaces. These parametrizations are not conformal, and no complex analysis is involved. If only someone had realized that these surfaces share the same Gauß map, the discovery of the Enneper-Weierstraß representation could have happened decades earlier.

Equation 16 is a mystery to me, I couldn’t verify that it satisfies the minimal surface equation.

Equation 20, Scherk’s fourth surface, is also quite complicated, but one of the components of the implicitly given surface does satisfy the minimal surface equation.

Using

t = 4\sin(x/2)^2+y^2\cos(x)\quad\text{and}\quad \rho^2 = t^2 + y^4 \sin(x)^2

the equation reads (slightly modernized)

\cosh\left( z+\sqrt{(t+\rho)/2} \csc(x/2)\right) = \frac{4 \sin(x/2)^2 + \rho}{y^2}

To find its Enneper-Weierstraß representation and make a decent image, I looked at the level curve for x=π, which simplifies to

1+\cosh\left(\sqrt{4-y^2}\right) = \frac{8}{y^2} \ .

This turns out to be a symmetry curve of the surface, so its normal lies in the plane x=0, and the Schwarz-Björling formula can be used to find the  Enneper-Weierstraß representation:

G(z) =\frac{z-1}{z+1} \quad\text{and}\quad dh = i\frac{z}{z^4-1} \ .

From here we can see that the surface is singly periodic with two annular and two helicoidal ends, and is also singular (at the points corresponding to 0 and infinity).

piece

Above you can see one half of the surface, with (parts of) both helicoidal ends and one of the annular ends. The singular point is where the horizontal symmetry curve in the middle meets the intersection of the two helicoidal ends, which is a straight line on the surface. Rotating about it gives a fundamental piece; below are three copies of it.

copies

For details, see the notebook under the resource below.

Amusingly, there is a simpler surface with the same type of ends that I accidentally discovered a while ago.

Finally, there is equation 30, giving the orthogonal case of Scherk’s singly periodic surface. Scherk does note some similarities to his doubly periodic surface.

Resources

Mathematica Notebook for Scherk IV