  We study smooth, proper embeddings of noncompact surfaces in 4-manifolds,
focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to
the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two
uncountable classes of exotic planes, with radically different properties. One
class is simple enough that we exhibit explicit level diagrams of them without
2-handles. Diagrams from the other class seem intractable to draw, and require
infinitely many 2-handles. We show that every compact surface embedded rel
nonempty boundary in the 4-ball has interior pairwise homeomorphic to
infinitely many smooth, proper embeddings in R^4. We also see that the
almost-smooth, compact, embedded surfaces produced in 4-manifolds by Freedman
theory must have singularities requiring infinitely many local minima in their
radial functions. We construct exotic planes with uncountable group actions
injecting into the pairwise mapping class group. This work raises many
questions, some of which we list.
