The Costa Surface

In his 1982 PhD thesis, Celso José da Costa wrote down the Enneper-Weierstraß representation of a complete minimal torus with two catenoidal and one planar ends, all with limiting vertical normals. 

I do not know whether Costa had any hope or even opinion that his surface might be embedded, but this is what David Hoffman and William Meeks realized and proved in 1985. It was the first complete, embedded minimal surface of finite topology after 1776 when Meusnier had proved that the helicoid is minimal. This breakthrough has spawned a vast number of new examples and triggered ongoing research. 

David Hoffman and William Meeks found more symmetric examples of higher genus and were also able to deform the middle planar end into a catenoidal end.

Putting the new surfaces under some regime of classification has proven difficult. Costa’s proof that 3-ended embedded minimal tori belong to the Costa-Hoffman-Meeks family is all but transparent, and the question whether there are other embedded minimal tori of finite total curvature is still open. Examples with more ends seem to require also more handles, like Meinhard Wohlgemuth’s examples.

Then there are periodic examples that utilizes Costa saddles as building blocks, like the singly periodic Callahan-Hoffman-Meeks surface and the singly periodic Costa-Scherk surface below to the right that is different but possibly related to the Batista-Martín surface (of which I haven’t made a picture yet).

Below are two doubly periodic surfaces where the Costa saddles are rotated by 45º. The left one is the Lübeck-Batista surface, the right one a doubly periodic Callahan-Hoffman-Meeks surface with reflectional symmetries and without straight lines.  Can one rotate a Costa saddle continuously by 360º in any such configuration?

Finally, there are several triply periodic Costa surfaces. The left is Alan Schoen’s I6 surface from around 1970, found through soap film experiments, and predating the Costa surface by over 10 years. The middle one is Batista’s surface, and the right one a new example of genus 4 that actually has the Costa surface as a limit, and not the Callahan-Hoffman-Meeks surfaces.

All this is only a beginning. Laurent Hauswirth and Frank Pacard have smuggled a Costa saddle into Riemann’s minimal surface, making it a genus one surface with infinitely many ends. Laurent Hauswirth has also used Costa saddles to construct families of singly periodic surfaces with annular ends.


Mathematicians like to classify things. Among the complete, embedded minimal surfaces of finite total curvature in Euclidean space or space forms, this has been accomplished in most of the simplest possible cases. Let’s summarize:

In Euclidean space there are only two surfaces of genus 0 in this class: The plane and the catenoid. This is a consequence of the López-Ros theorem, proven in 1991 by Francisco López and Antonio Ros.

For translation invariant surfaces (or equivalently, minimal surfaces in ℝ³ divided by a translation, the surfaces of genus 0 (in the quotient) are the general Karcher-Scherk saddle towers. These surfaces have an even number 2n of annular ends and 2n-3 free parameters with which they can flap their ends. This has been proven by Joaquín Pérez and Martin Traizet in 2007.

If you want planar ends, the lowest possible genus is 1, and Bill Meeks, Joaquín Pérez and Antonio Ros have shown in 1998 that the Riemann minimal surfaces are the only ones.

The situation is not yet resolved for the screw motion invariant surfaces. Conjecturally, these surfaces should be Hermann Karcher’s twisted saddle towers.

The case of doubly periodic surfaces of genus 0 has been settled by Bill Meeks and Hippolyte Lazard-Holly in 2001. These surfaces have non-parallel top and bottom ends.

Doubly periodic surfaces with parallel top and bottom ends can only occur in genus one and higher. Again, the genus one case has been classified: Joaquín Pérez, Magdalena Rodríguez and Martin Traizet have shown in 2005 that these are the KMR surfaces.

The main open question is that of a classification of triply periodic minimal surfaces of genus 3. To describe the state of the art will deserve several dedicated blog posts.

Likewise, I will outline in future posts the state of the art in the next difficult (open) cases.

Finally, I should mention that there are other, equally valid viewpoints for classification, using different assumptions about the topology.


F.J. López and A. Ros, On embedded complete minimal surfaces of genus zero, Journal of Differential Geometry 33 (199), 293–300

J. Pérez, M. Traizet: The Classification of Singly Periodic Minimal Surfaces with Genus Zero and Scherk-Type Ends, Transactions of the American Mathematical Society
359 (2007), 965-990.

W. H. Meeks III, J. Pérez, A. Ros: Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998),107–132

H. Lazard-Holly and W. Meeks: Classification of doubly-periodic minimal surfaces of genus zero, Invent. math. 143 (2001), 1–27.

Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), 523–577

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.

Printing Scherk in Clay

Having a virtual repository is great because it is widely available and doesn’t require space. Sometimes, however, one likes to be able to look at something real, so occasionally I will post about actual objects involving minimal surfaces.


Two years ago, Malcolm Mobutu Smith and myself set out to make mathematically inspired objects in clay. Malcolm has been intrigued by the relatively new method of clay printing, so he built a small printer, and we got to work.


The simplicity is challenging: You have a tube full of clay that is providing a continuous stream of clay (unless there are air pockets in the tube), a little motor that moves the tube around horizontally and vertically (don’t stop, unless you really want a small heap of clay), and a little Arduino to whom you can talk in Gcode.

My little mesh.m package has a function that allows to thicken a surface mesh, which can be exported into an stl file. Then we use Slic3R to convert that into Gcode, which is not much more than a bunch of instructions saying “move from A to B in time T.”


After that, the printing of a 6-ended singly periodic Scherk surface starts with layers of three arcs. Because the printer doesn’t stop printing, it needs to skip fast between the arcs, leaving behind little charming artifacts.

Many things can go wrong: Overhangs can (will) break, the clay dries too fast so that the next layer doesn’t stick, the clay is too soft so that everything sags… But this piece worked out pretty nicely.  I now have a real Scherk surface at home:


Three Ends (Parametrizations I)

Most computer algebra systems come with some capabilities to render parametrized surfaces in space. You usually specify three functions of two variables x and y and a rectangle in the (x,y)-plane, and are rewarded with an image.


This has limitations: The most complicated topology you can achieve this way is a torus. Things get tricky when you want to draw something that has more than two ends.

Besides being able to draw these surfaces at all, one would also like to use a conformal parametrization so that the images of the parameter lines become orthogonal in space. This helps us being illusioned, because, having grown up in environments full of right angles, we assume that any intersection happens at a right angle.


This can be accomplished for 3-ended surfaces by moving the ends to -1, 1 and infinity (using a Möbius transformation), dividing the plane into quadrants, and mapping a rectangle to the first quadrant so that we get polar coordinates at 1 and infinity as shown above. This is done using

f(z) = \sqrt{e^z+1}

on a rectangle of the form [a,b] x [0,π]: The exponential function maps the rectangle to a half-annulus in the upper half plane centered at 0. We then shift the “hole” at 0 to 1 and take a square root which bends the 180º angle at 0 to a right angle. The only thing to remember is that we want to have a parameter line hitting the origin, because otherwise our parameter mesh will have a gap there.

This is one of the simpler explicit parametrizations and responsible for the images on this page.

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.


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.


Mathematica Notebook for Scherk IV