Criterion for deciding the conformal class of a metric on a complete surface

A Riemann surface that is topologically a disc is conformally equivalent to the plane if and only if its boundary is a puncture. That is, there is a neighbourhood of the boundary that is conformally equivalent to the punctured disc. Equivalently, and more geometrically, for any compact set K the family of curves separating infinity from K has infinite modulus.

The equivalent version with two punctures works for the punctured plane / cylinder.

(Note that this is essentially an extended version of the comment made by retract-subpolyhedron.)

This is a classical problem which is called the Type Problem of a simply connected Riemann surface: If you have a metric on an open simply connected surface, to determine whether it is conformally equivalent to the plane or to the disk.

The general situation is the following: there are necessary and sufficient conditions, but they are usually difficult to verify for specific metrics. And there are very many separate necessary or sufficient conditions which are easier to verify, especially for some special classes of metrics. There was an intensive research on this in 1930-1950, and a few modern papers.

Some references are:

MR2019938 Benjamini, Itai; Merenkov, Sergei; Schramm, Oded, A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature. Proc. Amer. Math. Soc. 132 (2004), no. 3, 641–647,

MR0954627 Doyle, Peter G. On deciding whether a surface is parabolic or hyperbolic. Geometry of random motion (Ithaca, N.Y., 1987), 41–48, Contemp. Math., 73, Amer. Math. Soc., Providence, RI, 1988.

MR0428232 Milnor, John On deciding whether a surface is parabolic or hyperbolic. Amer. Math. Monthly 84 (1977), no. 1, 43–46.

MR0279280 Nevanlinna, Rolf Analytic functions. New York-Berlin 1970 viii+373 pp.

MR0049330 Volkovyskiĭ, L. I. Investigation of the type problem for a simply connected Riemann surface. (Russian) Trudy Mat. Inst. Steklov. 34, (1950). 171 pp. This is the most comprehensive exposition. Not much was added to this theory since 1950.