Mathematicians like to classify things. Among the complete, embedded minimal surfaces of finite total curvature in Euclidean space or space forms, this has been accomplished in most of the simplest possible cases. Let’s summarize:
In Euclidean space there are only two surfaces of genus 0 in this class: The plane and the catenoid. This is a consequence of the López-Ros theorem, proven in 1991 by Francisco López and Antonio Ros.
For translation invariant surfaces (or equivalently, minimal surfaces in ℝ³ divided by a translation, the surfaces of genus 0 (in the quotient) are the general Karcher-Scherk saddle towers. These surfaces have an even number 2n of annular ends and 2n-3 free parameters with which they can flap their ends. This has been proven by Joaquín Pérez and Martin Traizet in 2007.
If you want planar ends, the lowest possible genus is 1, and Bill Meeks, Joaquín Pérez and Antonio Ros have shown in 1998 that the Riemann minimal surfaces are the only ones.
The situation is not yet resolved for the screw motion invariant surfaces. Conjecturally, these surfaces should be Hermann Karcher’s twisted saddle towers.
The case of doubly periodic surfaces of genus 0 has been settled by Bill Meeks and Hippolyte Lazard-Holly in 2001. These surfaces have non-parallel top and bottom ends.
Doubly periodic surfaces with parallel top and bottom ends can only occur in genus one and higher. Again, the genus one case has been classified: Joaquín Pérez, Magdalena Rodríguez and Martin Traizet have shown in 2005 that these are the KMR surfaces.
The main open question is that of a classification of triply periodic minimal surfaces of genus 3. To describe the state of the art will deserve several dedicated blog posts.
Likewise, I will outline in future posts the state of the art in the next difficult (open) cases.
Finally, I should mention that there are other, equally valid viewpoints for classification, using different assumptions about the topology.
F.J. López and A. Ros, On embedded complete minimal surfaces of genus zero, Journal of Differential Geometry 33 (199), 293–300
J. Pérez, M. Traizet: The Classification of Singly Periodic Minimal Surfaces with Genus Zero and Scherk-Type Ends, Transactions of the American Mathematical Society
359 (2007), 965-990.
W. H. Meeks III, J. Pérez, A. Ros: Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998),107–132
H. Lazard-Holly and W. Meeks: Classification of doubly-periodic minimal surfaces of genus zero, Invent. math. 143 (2001), 1–27.
Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), 523–577