Costa's minimal surface

Example

Costa's surface is a complete minimal surface in $\R^3$ that is embedded and has finite total curvature. Topologically it is a torus with three points removed. Near the three removed points the surface has three ends: two look like the ends of a catenoid and one looks like a plane.

It is built with the Weierstrass–Enneper parameterization. The domain is the square torus $\C / (\Z + i\Z)$ minus three points, and the data are given by the Weierstrass $\wp$ function of this lattice. One checks that the real periods of the integrals vanish, so the surface is well defined, and minimality then follows from the parameterization. The hard part, proved later, is that the surface has no self-intersections.