On the Existence of a Proper Minimal Surface in R with a Conformal Type of Disk

Proper minimal surfaces in R3 have peculiar properties that are not shared by general minimal surfaces; especially in the embedded case. It has been proved that, under additional conditions, this family of surfaces has strong restrictions on their conformal structures. For instance, Huber and Osserman proved that if M is a complete minimal surface with finite total curvature, then M has the conformal type of a compact Riemann surface minus a finite number of points. In particular it is parabolic, that is, M is not compact and M does not carry a negative non-constant subharmonic function. In the same context, Collin, Kusner, Meeks, and Rosenberg [CKMR] have proved that if M is a properly immersed minimal surface in R3, then M(+) = {(x1, x2, x3) ∈ M : x3 ≥ 0} is parabolic. Furthermore, López [L] have shown that a properly immersed minimal surface with compact boundary and finite topology omitting some special subset of R3 is parabolic. This kind of results have motivated the following conjecture: Conjecture (Meeks, Sullivan [Me]). If f : M → R3 is a complete proper minimal immersion where M is a Riemannian surface without boundary and with finite genus, then M is parabolic.