  We present a new connection between the Hele-Shaw flow, also known as
two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with
boundary contained in a totally real submanifold. Using this we prove short
time existence and uniqueness of the Hele-Shaw flow with varying permeability
both when starting from a single point and also starting from a smooth Jordan
domain. Applying the same ideas we prove that the moduli space of smooth
quadrature domains is a smooth manifold whose dimension we also calculate, and
we give a local existence theorem for the inverse potential problem in the
plane.
