  A thin water film evaporating from a cleaved mica substrate undergoes a
first-order phase transition between two values of film thickness. During
evaporation, the interface between the two phases develops a fingering
instability similar to that observed in the Saffman-Taylor problem. The
dynamics of the droplet interface is dictated by an infinite number of
conserved quantities: all harmonic moments decay exponentially at the same
rate. A typical scenario is the nucleation of a dry patch within the droplet
domain. We construct solutions of this problem and analyze the toplogical
transition occuring when the boundary of the dry patch meets the outer
boundary. We show a duality between Laplacian growth and evaporation, and
utilize it to explain the behaviour near the transition. We construct a family
of problems for which evaporation and Laplacian growth are limiting cases and
show that a necessary condition for a smooth topological transition, in this
family, is that all boundaries share the same pressure.
