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:

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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.

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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.

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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.

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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.