Winding Numbers

In 1960, Robert Osserman  proved that a complete minimal surface of finite total curvature is conformally a compact Riemann surface with finitely many points removed, and the Enneper-Weierstraß representation extends meromorphically to the punctures.

One could now attach to any such surface a number of invariants: the genus g of the surface, the degree deg G of the Gauss map, the number e of ends, and for each end a winding number \nu_j. The latter is computed by subtracting 1 from the maximal order of the poles of the Weierstraß 1-forms at that end. Geometrically, small circles about the puncture of shrinking radius are mapped to space curve that can be rescaled so that they  converge to a circle with that winding number as multiplicity.

Fritz Gackstatter (1976) and independently Luquesio Jorge and William Meeks (1983) proved a useful winding number formula for oriented minimal surfaces of finite total curvature:

2 deg G = 2g-2 +\sum_{j=1}^e \nu_j+1

For instance, the catenoid has genus 0, the degree of the Gauss map is 1, and there are two ends of winding number 2. Likewise, the the Enneper surface has genus 0, the degree of the Gauss map is 1, and there is one of winding number 3.  These are, as Osserman proved, the only complete minimal surfaces with total curvature -4π.

cg-1

The next case of total curvature -8π was treated by F. López. Most prominently in his list is the Chen-Gackstatter surface, the only minimal torus of total curvature -8π.

Besides that, there are numerous spheres. One can have (by the winding number formula) one end of winding number 5, or two ends with winding numbers 1 and 3 or 2 and 2, or three ends with winding number 1 each. You find examples for all cases somewhere on this page.

Here is a question I don’t know the answer to: Can one have a complete minimal surface of finite total curvature with just one end of winding number 2? At first, this appears to contradict the winding number formula due to parity, but the surface could be non-orientable, like F. López’ amazing minimal Klein Bottle (which has a single Enneper end with winding number 3).

bottle

Exemplum VII

In 1744, Leonhard Euler published a book with the succinct title Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In it, he develops a general method to find curves that satisfy extremal problem, which is cow called the Calculus of Variations. In contrast to the ordinary calculus which allows to find extrema of a single function by solving an equation involving the derivative of a function, here a functional is minimized or maximized over all functions by solving an ordinary differential equation.

Catenoid-far

His example VII has the title Invenire curvam, qua, inter omnes alias ejusdem longitudinis, circa axem AZ rotata, producat solidum superficies fit vel maxima vel minima.

Euler’s Latin almost doesn’t require a translation into English: To find a curve, which among all others with the same length (meaning defined over the same interval) and rotated about the z-axis, produces a solid whose surface shall be maximal or minimal.

Euler then proceeds, in a few lines, to apply his method to derive the differential equation for finding curves so that the corresponding surface of revolution has extremal area. Euler notes that this equation is solved by the catenary.

I am not a historian, so I do not know who coined the term catenoid, nor do I know who made a first image.

Euler  is not concerned with two catenaries passing through the same points and thus offering two different solutions of evidently different area.

TwoCatenoids-1.7

Euler neither discusses nor defines the term minimal surface. This is done 1760 by Joseph Lagrange, who establishes in his note Essai d’une nouvelle methode pour determiner les maxima et les minima des formules intégrales indéfinies the minimal surface equation for a graph, observes that planar graphs satisfy his equation, and adds that “la solution générale doit être telle, que le périmètre de la surface puisse être détermine a volonté”  –the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily. Lagrange gives no further examples, but his comment has triggered research that is still ongoing.

 

 

 

Mission Statement

The purpose of this repository is to provide annotated high quality images, animations, and 3D data  of minimal surfaces.

It will consist mainly of two components: The repository that organizes the known surfaces and provides images, data, and references, and a blog that provides context, makes connections, explains things and tells anecdotes.

At the moment, there is very little here, but this will change rapidly. Bear with me while I struggle with WordPress. I plan to add a weekly blog post, and add content to the repository at a fast pace.

The main purpose, however, is to make the known minimal surfaces available to a broad community, including researchers, artists, and other interested people. To facilitate this, all the material can be used under the Creative Commons license below.

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This project has been made possible through  the very generous support of a donor who wishes to remain unnamed.