On embedded complete minimal surfaces of genus zero

Properly embedded minimal surfaces with finite topology

“…The simplest examples in this family can be characterized in terms of its topology. If the surface has more than one end, the above result was proved by López and Ros [26]. In the one ended case this is a recent result of Meeks and Rosenberg [38].…”

“…Under suitable global assumptions this deformation gives strong restrictions on the geometry and the topology of minimal surfaces all of whose fluxes are vertical, see works of López, Pérez and Ros [26,47,54,44]. In particular we have the following result.…”

“…In the case k = 0, this is has been proved in [26]. For higher genus, Meeks, Pérez and Ros [34] have proved that, given k, M(k, r) is empty for r large enough.…”

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

“…The first topological obstructions for complete embedded minimal surfaces M of finite total curvature were given by Jorge and Meeks [20], who proved that if M has genus zero, then M does not have 3, 4 or 5 ends. Later this result was generalized by López and Ros [22] who proved that the plane and the catenoid are the only genus zero minimal surfaces of finite total curvature in M. About the same time, Schoen [37] proved that a complete embedded minimal surface of finite total curvature and two ends must be a catenoid.…”

Complete Embedded Minimal Surfaces of Finite Total Curvature

“…(Theorems 3.1 and 3.4, due to Lopez-Ros [51], Schoen [67] and Costa [16].) We present a proof (in Section 3.1) of the Lopez-Ros theorem, which states that a complete minimal surface of genus zero and finite total curvature must be the plane or the catenoid.…”