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

Eccentricity

In his thesis, Peter Connor discusses doubly periodic minimal surfaces with parallel top and bottom ends that are cut by vertical planes into simply connected pieces. Here are two examples:

These surfaces can be systematically described using polygonal domains like the one below.

The left and right vertical edges where the curves end correspond to the ends of the surface, and the corners to the points with vertical Gauss map, recording whether it points up or down by a left or right turn. There are only finitely many possibilities for each genus, and they are easy to enumerate. What is not so easy is to find out whether the corresponding surfaces actually exist, either numerically or theoretically. For genus 3 and higher, existence proofs usually rely on understanding the limits of these surfaces, and this is what this post is about. When approaching a limit, a minimal surface necessarily decomposes into fundamentally simpler surfaces.

Above are two examples of Connor’s experimental doubly periodic surfaces of genus 3, near one of their limits. On the left hand side we see 8-ended singly periodic Scherk surfaces emerging, on the right hand side 10-ended Scherk surfaces. There is a simple count that helps to predict what can arise. This count is based on the fact that the total curvature of these surfaces is an integral multiple of 4π, and this multiple stays the same while deforming and approaching limit. Let’s call this multiple the degree of the surface (it is, in fact, nothing but the degree of its Gauss map). The Catenoid and the single and doubly periodic classical Scherk surfaces have all degree 1. More generally, a 2n-ended singly periodic Scherk surface of genus 0 has degree n-1, and a doubly periodic surface with four ends and of genus g has degree g+1. So the two surfaces above have all degree 4. For the left, the limit is a singly periodic Scherk surface of degree 3, but the components are stitched together using a catenoidal neck, adding 1 to the degree. The surface on the right limits in a Scherk surface of degree 4, so doesn’t need stitches.

Above are two more examples of genus 3 from Connor’s list. The left decomposes into two doubly periodic Karcher-Scherk surfaces of degree 2, the right one into two KMR surfaces, which have degree 2 each.

The big surprise of Connor’s series of examples of genus 3 was the surface to the left. It has the same combinatorial polygon description is the (known) RTW-surface to the right, but the handle has become eccentric. The limit in either case is a toroidal Scherk surface with 8 ends, which has degree 4. In the right case, this is again a known surface, but the left case is new. Proving the existence of either the doubly or the singly periodic version would be highly desirable.

A Tale of Two Squares

Today, we are pushing things a little. We begin with a minimal heptagon with two horizontal straight edges, and the other five edges lying in symmetry planes parallel to the vertical coordinate planes. We also want this heptagon arranged so that extension by reflections/rotations creates an embedded triply periodic surface.

Above is a first example, showing a translational fundamental piece of a triply periodic surface of genus 6 that is tiled by eight such heptagons. We see that the horizontal straight segments are replicated into edges of squares, so that all surfaces we will look at today will sit one way or another between two squares.

A way to encode the possibilities for such heptagons is through the picture above. It represents the divisor of the square of the Gauss map on the quotient torus under the order 2 rotation about a vertical axis. Less technically, the grayed rectangle represents the heptagon, with the two vertical blue edges corresponding to the horizontal straight segments. The red dots are the seven heptagon vertices, and the symbols 0 and ∞ tell whether the Gauss map points up or down.

There are 12 possibilities that lead to different surface candidates. For each of these, a 3-dimensional period problem needs to be solved, leading in the case of success to 1-parameter families. In 8 of these cases, I was able to solve this period problem numerically. No existence or non-existence proof is known in any of these cases at this point.

There are several motivations behind this exploration. For one, I would like to know what limits can occur, and how the limit surfaces are combined. This suggests possibilities for general gluing constructions that would establish the existence of these and many other surfaces, Secondly, some of the limits are surfaces that are either only very difficult to obtain (like the Callahan-Hoffman-Meeks surface of higher genus that emerges above), or have also only been established numerically (like the exotic doubly periodic Karcher-Scherk surface of genus 3 below).

Finally, there is always the chance that an exhaustive enumeration leads to examples with unexpected properties, like the surface below, that in a diagonal view reveals an appealing tunnel system that indicates that we are probably having a special pair of skeletal graphs in front of us.

Resources

Main Page for all 8 surfaces

To Be or Not To Be

Existence is an interesting concept. Even in the most rigorous of sciences, it is not free of dispute.

Above is an image of the Horgan surface, a minimal surface that does not exist, at least not with all the glorious property it appears to have. There are less convincing attempts to convey the existence of a non-existing object, like the catenoid with a handle:

In Mathematics, pictures provide evidence and enjoyment. Existence or non-existence requires proof, and provides certainty. Still, the membrane between to be and not to be can be very thin, as in the example at the top.

The pattern made by the catenoidal necks in the top image – two up and two down, at the corners of a square – does actually occur. One of the first examples I have seen is the Callahan-Hoffman-Meeks surface with an added handle as above. Its existence is not so easy to prove – does it therefore exist a little less?

The triply periodic surface above is the simplest version of the Catenoid-Square pattern I know of, and much more recent. So, in a sense, all this research is to an unconfessed extent concerned with probing the border of existence, from either side.

The Upper Half Disk (Parametrizations 2)

One of the text book examples of conformal maps is given by the complex function f(z) = z+1/z which maps the upper half disk bijectively to the lower half plane. This is the main ingredient for the following problem:

We would like to have a conformal map from a rectangle to a half disk (say of radius r) so that the left and right sides of the rectangle are mapped to half circular arcs around two points a and b.

The inverse of this map can be obtained by composing the text book map above with a Möbius transformation that sends the images of a and b to 0 and infinity, and then use polar coordinates on the upper half plane. Details are in this notebook.

What is this good for? Examples are 4-ended surfaces like the ones above that have sufficient symmetry that one can place the ends as four punctures on the real line, symmetric with respect to a circle.

This can also be useful more generally when existing symmetries cut the surface into pieces with four ends along the boundary of the pieces, as above.

Out of the Box

One of many ways to construct triply periodic minimal surfaces is via conjugate surfaces. You start with a polygon in space, solve the Plateau problem, construct the conjugate surface. Instead of straight edges, this conjugate surface piece will allow extension by reflecting across its boundary edges, and with a fair amount of luck, you obtain a nice triply periodic surface. There are two disadvantages of the Plateau method: You are numerically limited to minimizing Plateau solutions, and the whole approach gives little theoretical insight. Here is a variation of this approach:

We start with a minimal polygon inside a box with all edges perpendicular to the faces of the box. Reflections at the faces will produce 8 copies, which constitute a translational fundamental piece of a triply periodic surface. If we look at the boundary of the polygon in the vertical faces, we note that at the corners the Gauss map will be vertical. We encode this in a sequence of + and – signs. For the left boundary component in the example, we have two points with normal pointing (say) up, encoded by +. In the second component we first point down at the upper point, and then down at the lower point, encoded by +-. Both sequences give the symbol (++|+-).

The same information is also contained in the shaded rectangle above, with the red dots labeled a and b corresponding to the corners in the left boundary edge, and c and corresponding to those in the right edge. The entire rectangle then represents the torus quotient of the surface under the 180º rotation about the vertical axis. The vertices are the zeroes and poles of the Gauss map.

Together with the additional reflectional symmetries at the horizontal box faces (the vertical green lines in the rectangle), this information determines the Gauss map. The height differential on the quotient torus is just dz, so we have the entire Weierstrass representation of the surface, except that we do not know the values of the parameters a,b,c,d and τ.

A linear combination of the parameters a,b,c,d determines how the Gauss map rotates in horizontal symmetry planes. For 8-gons as above, one usually is then left with a 2-dimensional period problem, resulting in a 2-dimensional family of examples. This approach is useful for three reasons: One can use the Enneper-Weierstrass representation for theoretical and numerical purposes, investigate limits easily, and extend the method by forsaking the horizontal symmetries, as we will see at a later point.

This page shows examples for these seven types, you can find more under the individual surface pages, listed under the genus 5 box types section in the triply periodic minimal surfaces page.

Meeks vs Non-Meeks

One of the most active areas of minimal surface research concerns embedded triply periodic minimal surfaces. They are periodic with respect to a 3-dimensional lattice of translations, and roughly classified by the genus of the quotient surface. The simplest case is that of genus 3, and even that is far from being understood. The earliest set of examples is due to Hermann Amandus Schwarz, and he already understood well that they come in parameter families. Below you can see the P surface, the D-surface, and members of the H and CLP-families.

So, how many are there? A casual parameter count tells that if such a surface is reasonably generic, it should come in a smooth 5-parameter family, not counting rotations, translations, and scalings. Five is a fairly large number. The space of possible lattices is also 5-dimensional, and in the best of all possible worlds, there would be precisely one surface in every lattice. Unfortunately, it is not even true that small surface deformations are in 1:1 correspondence to small lattice deformations, and things are dramatically more complicated.

Alan Schoen added to the Schwarz examples the Gyroid (left) a member of the associate family of the P surface, and Sven Lidin the Lidinoid (right), an associate member of the associate family of a particular H surface.

Key in their construction was an understanding of the Gauss map: This is a map of degree 2 to the sphere, branched over 8 points. Schwarz had already realized their significance; above is a reproduction of Plate VI from his 1871 monograph. Figures 43 and 44 show the branched values of the oP and tP deformations of the P surface, figure 45 of the CLP surfaces, figure 46 represents a special case of the rPD family, and figure 47 the H family.

Bill Meeks realized in 1970 that for any choice of 8 branched values of the sphere that is antipodally symmetric, there are two (conjugate) embedded triply periodic minimal surfaces with these branched values. This Meeks-family covers all then known examples, except for the H-surfaces, the Gyroid, and the Lidinoid.

Since then, progress has been made in small steps. Above are two new Non-Meeks surfaces (called and oH) due to Hao Chen and myself that can be deformed into different Meeks surfaces. It is not clear at this point whether they are related. Hao Chen has also recently proven that the Lidinoid and Gyroid belong to a 1-parameter family. I am sure this year will see more progress.

Here are two immediate questions that I would love to have an answer for:

  • For what triply periodic minimal surfaces are there embedded surfaces in the associate family that are not conjugates? For genus 3, the only examples so far are the Lidinoid and the Gyroid.
  • We know that the H-surfaces belong to a 5-dimensional family (work of Martin Traizet) that intersects the Meeks family (work of Hao Chen and Matthias Weber). Are the Gyroid/Lidinoid part of this family, or do they belong to a third, separate family?