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 . 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:

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.

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).

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