Ends of Minimal Surfaces 



The ends of complete minimal surfaces can be as complicated as you wish. There are two reasonable assumptions which put severe restrictions on their shape:  
The finite total curvature condition implies that the Weierstraß data of the
minimal surface extend meromorphically to a compact Riemann surface. This means essentially
that the ends cannot be much worse then the ends of an Enneper surface. To the right you see a minimal sphere with two such Enneper ends. 

Only two finite total curvature ends are embedded: The catenoid end and the planar end. To the right is a surface with two catenoid and one planar end. All ends are embedded but the planar end will eventually intersect the catenoid end.  
Here you can see a sphere with six catenoid ends. There is an abundance of nonembedded finite total curvature surfaces.  
Under the assumption that the surface is embedded and of finite topology, there are two possibilities (Collin): Either the surface has finite total curvature, or it has just one end which is asymptotic to the helicoid (Hauswirth/Perez/Romon). The latter can occur, as the genus one helicoid ilustrates. To the right you can see a piece of a minimal surface with an exotic helicoidal end. 