I've only just heard of Costa's minimal surface. I'm interested in the fact that it may be formed by puncturing a compact surface.
Another, more simple, minimal surface is the Catenoid. This also may be formed by puncturing a compact surface. In three dimensions this is just a "handle" added to the 3-sphere, and has the topology $$S^{2}\times S^{1}$$
Can the topology of Costa's minimal surface puncturing the torus be written in a similar fashion, for example in dimension 3? I was thinking it could be as simple as $S^{1}\times S^{1}\times S^{1}$???