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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.

From the Kähler-Ricci flow to moving free boundaries and shocks