  We study classical solutions to the one-phase free boundary problem in which
the free boundary consists of smooth curves and the components of the positive
phase are simply-connected. We show that if two components of the free boundary
are close, then the solution locally resembles an entire solution discovered by
Hauswirth, H\'elein and Pacard, whose free boundary has the shape of a double
hairpin. Our results are analogous to theorems of Colding and Minicozzi
characterizing embedded minimal annuli, and a direct connection between our
theorems and theirs can be made using a correspondence due to Traizet.
