Björling Surfaces I

Plateau’s problem asks to find a minimal surface that spans a given closed curve. This is a global question, and the answers are delicate. In contrast, there is a much simpler local problem, posed by Emanuel Gabriel Björling in 1844:

For a given space curve and normal field along the curve, find a minimal surfaces that contains the curve and has the given normal as surface normal. For instance, for a circle with outer normal we expect a catenoid, and for a straight line with a normal that rotates with constant speed a helicoid.

In his 1844 paper, Björling proved that for any given real analytic space curve and real analytic normal field along the curve, there is a unique local solution. Later, Hermann Amandus Schwarz gave an explicit formula for the Weierstrass data of the solution.

The fact that the solution of Björling’s problem is rather simple has been quite fertile. For instance, one can let a normal rotate about a circle and get Pablo Mira’s circular helicoids. However, one quickly finds oneself in the default situation of the wizard’s apprentice who has learned how to use spell but not acquired knowledge about the consequences.

For instance, who would have thought that if one starts with a planar cycloid as a curve and takes as normal the normal vector to the curve, the resulting surface (already known to Eugène Charles Catalan) contains parabolas as geodesics? Also, the formulas that Schwarz gave us are not always easy to integrate. The helicoids winding along logarithmic helices below (due to Christine Breiner and Stephen Kleene) can be explicitly described, but the formulas span an entire page.

So it would be desirable to have some control about the global nature of the solutions, and also some insight when to expect explicit formulas. We will earn about this next week.

Resources

E.-G. Björling: In integrationem aequationis Derivatarum partialium superficiei, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Archiv der Mathematik IV (1844), 290-315

Mathematica Notebook with Examples

Björling Surfaces Repository Page

Wrapped Packages

Another fascinating minimal surface from Alan Schoen’s NASA 1970 report is his I-WP surface.

The name I-WP indicates the two skeletal graphs of the complement: The I-graph and the WP-graph. WP stands for wrapped package. You can see 8 copies of the surface below.

Because it is cut by symmetry planes into simply connected pieces, the conjugate surface is tiles with minimal polygons. This is Steßmann’s surface, discovered 1931, about 40 years earlier.

Berthold Steßman’s thesis determined the Enneper-Weierstrass data of all minimal quadrilaterals such that rotations about the edges generate a discrete group, completing work begun by Riemann and Enneper another 70 years earlier. One might wonder why Schoen’s I-WP surface was not discovered much earlier. Likewise, one might wonder why Steßmann (and Carl Ludwig Siegel, his advisor in Frankfurt), was interested in the Enneper-Weierstrass representation when Jesse Douglas and Tibor Radó had established the existence of much more general Plateau solutions by 1931.

The reasons for progress or the lack of it often lie in human fate. I could find only little about Berthold Steßmann. A short biographical note by the German Mathematical Society mentions that he was born on August 4, 1906 in Hüllenberg, Germany, studied in Göttingen and Frankfurt to become a high school teacher, which he completed in 1933. Then, a year later, he received his PhD about periodic minimal surfaces, with Carl Ludwig Siegel as advisor. The note also mentions that Steßmann was Jewish. I have some hope that his doctoral degree helped him to emigrate in time. Another historical note mentions that he received the Golden Doctoral Certificate at the 50th anniversary of his doctorate in Frankfurt.

Such were the times: such are the times.

The story of I-WP continues a little further. Hermann Karcher found a tetragonal cousin which he called T-WP:

A final riddle: Sven Lidin, Stephen Hyde and Barry Ninham discovered that the associate family of the I-WP surface contains several embedded triply periodic minimal surfaces at angles that are multiples of 60º. These are, however, all isometric to the I-WP surface. It is conceivable however, that they possess different kinds of deformations.

Alan Schoen’s Nasa Report 1970

Alan Schoen celebrated his 94th birthday earlier this month, so it is only fitting to write a little about his NASA report in my series of blog posts of highly influential papers about the construction of minimal surfaces.

This task is not easy, because there is too much worth discussing, so I decided to split this into multiple posts, beginning today with his simplest surfaces. These are H’-T, H”-R, S’-S”, and T’-R’. I add to these P and H, found much earlier by Hermann Amandus Schwarz, because these six surfaces share enough properties so that a single piece of code can be used to compute all of them.

They all have a reflectional fundamental cell consisting of a right prism over a triangle, which must be of one of the types (3,3,3), (2,4,4), or (2,3,6), the numbers representing the triangle angles as fractions of 180º. Below is a piece of the T’-R’ surface in a (6,2,3) prism, and next to it how this prism fits into a translational fundamental piece.

If we remove from the skeleton of the prism the edges that intersect the surface, we obtain the two skeletal graphs of the surface, shown below in extreme wide angle perspective from above. That one of the graph has triangle layers and the other rhombic layers is the reason for Schoen’s naming convention: The letters T and R stand for triangle and rhombus.

The piece in a prism can also be used to effectively parametrize these surfaces. To do this, note that the vertical faces of the prism meet the surfaces in two arcs: One is a segment without corners, the other has two 30º corners where the arc switches from one face of the prism to another. These corners are also the only points with vertical normal. We can therefore conformally parametrize a surface piece in a prism by the shaded rectangle below.

The vertical edges correspond to horizontal symmetry lines, and the horizontal edges to symmetry lines in the vertical prism faces. In this way, the shaded rectangle corresponds to the flat structure of the height differential. The points marked a and -a correspond to the two corners, where the Gauss map as a pole and zero, respectively. The value of a for an (r,s,t)-prism is determined (a consequence of Abel’s theorem) as
$a = \frac12\frac{s}{s+t}$.
This has the curious consequence that the height of the corners of the surface piece are determined relative to the height of the prism. For instance, for T’-R’, the value of a is 1/5, meaning that if the prism has top and bottom face at height +1/2 and -1/2, then the two corners are at height +1/5 and -1/5.

The integrals of the Enneper-Weierstrass forms G dh and 1/G dh become Schwarz-Christoffel maps that map the horizontal gray strip to a periodic polygon or freeze pattern. For T’-R’ this looks like

For other surfaces in this group, the angles will change. The Schwarz P-surface corresponds to the (2,4,4) prism, a will be 1/4, and the frieze pattern has an additional symmetry:

The red lines correspond the the horizontal straight lines on the P-surface. We will see in a later post that the same method can be used to generate many more surfaces.

Finally, all the surfaces in today’s group come in a 1-parameter family and have similar limits. The catenoidal neck pattern and the singly periodic Scherk surface arrangement is encoded by the skeletal graphs:

Resources

Universal Mathematica Notebook

Derived from Scherk’s Examples

During my last semester as an undergraduate student at the Technical University in Berlin in 1984, Dirk Ferus mentioned in his Algebraic Topology class that there would be a geometry conference over the weekend, which he recommended attending. Stupid me, I didn’t go. I could have met my future advisor (Hermann Karcher), and I could have seen a future collaborator (David Hoffman) present the first images of the Costa surface.

This conference is also mentioned in the introduction of another paper from my list of highly influential papers with new examples of minimal surfaces, namely Hermann Karcher’s 1988 Embedded Minimal Surfaces Derived from Scherk’s Examples.

During the academic year 1984/85, I had attended two semesters of Karcher’s Differential Geometry. At the end of the second term he announced that while the third semester would only be for those specializing in geometry, we all should come for the first two weeks, because he intended to spend them with explaining the basics about minimal surfaces, which he had completely neglected. I was a little disappointed, because I was eager to learn about the darker arts – symmetric spaces, Einstein manifolds, Finiteness Theorems…

Karcher didn’t just spend the first two weeks on minimal surfaces, but about half of the semester, developing and presenting what would become the paper mentioned above.

The images here represent only a selection of the surfaces described in that paper: There are the saddle towers, the toroidal half plane layers, and the helicoidal saddle towers. Besides all these example Karcher develops a method to derive the complex analytic Enneper-Weierstraß data from geometric features of the surface, which, ultimately, has led to the enormous zoo of examples we are dealing with today.

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.

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).

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.

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