  We show that the twisted K\"ahler-Ricci flow on a complex manifold X
converges to a flow of moving free boundaries, in a certain scaling limit. This
leads to a new phenomenon of singularity formation and topology change which
can be seen as a complex generalization of the extensively studied formation of
shocks in Hamilton-Jacobi equations and hyperbolic conservation laws (notably,
in the adhesion model in cosmology). In particular we show how to recover the
Hele-Shaw flow (Laplacian growth) of growing 2D domains from the Ricci flow. As
we briefly indicate the scaling limit in question arises as the
zero-temperature limit of a certain many particle system on X.
