Double Periodicity

Doublyscherk

Last week I explained a really complicated way to get from Scherk’s doubly periodic minimal surface to the helicoid, through a family of Schwarz Diamond surfaces. As was known already to Scherk, this can be done much easier, namely by “shearing” the standard Scherk surface above. I put apostrophes because a simple Euclidean shearing isn’t enough to keep the surface minimal.

Doublyscherkshear

Bill Meeks and Hippolyte Lazard-Holly have shown that these are the only embedded doubly periodic minimal surfaces of genus 0 (after taking the quotient by their translational periods). Things get tricky for larger genus. 

Scherk g=1

 

First of all one needs to distinguish whether the parallel half planes “on top” are parallel to the ones “at the bottom” or not. Today we stick with the case that they are not parallel, and are in fact orthogonal. Then there is just one such surface of genus 1 (I am pretty sure, but I think nobody has written a proof). This was first constructed by Hermann Karcher. It’s pretty clear (and provable) that one can continue like this, creating doubly periodic surfaces with more handles, like the genus 2 example below.

Scherk g=2

It would be a nice theorem if they all would be unique. But I don’t think so. Below is a picture of a genus 3 surface where the handles are arranged differently.

Scherk g=3 exotic

Proving that this really exists won’t be easy, but interesting, because it would allow one to speculate what will happen if one can shear this surface like the original Scherk surface.

 

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Scherk meets Enneper

My little excursions into the history of minimal surfaces continues with a contribution of Heinrich Scherk from 1835. Making assumptions that allowed him to separate variables in the so far intractable minimal surface equation, he was able to come up with several quite explicit solutions, two of which are still of relevance today.

Scherksimple

In its simplest version, the singly periodic Scherk surface looks from far away like two perpendicular planes whose line of intersection has been replaced by tunnels that alternate in direction.

The next milestone concerning these surfaces took place 1988, over 150 years later, when Hermann Karcher constructed astonishing variations. Among others, he showed they can be had with (many) more wings

Scherk9

and even twisted:

Scherktwist

Now, can they also be wiggled? The prototype here is the translation invariant Enneper surface. It has the feature that it can be wrapped onto itself after sliding it any distance.
In other words, it is continuously intrinsically translation invariant.

TransEnneper

Hmm. I should patent this.

So we can switch out the boring flat Scherk wings with the wiggly Enneper wings, like so, still keeping everything minimal, pushing the notion to its limits.

Enneper scherk1

Here is a more radical version. You don’t want to run into this in the wild.

Enneper scherk2