Competitive Hele-Shaw flow and quadratic differentials

Abstract

We introduce and investigate a generalization of the Hele-Shaw flow with injection where several droplets compete for space as they try to expand due to internal pressure while still preserving their topology. Droplets are described by their closed non-crossing interface curves in $\mathbb{C}$ or more generally in a Riemann surface of finite type. Our main focus is on stationary solutions which we show correspond to the critical vertical trajectories of a particular quadratic differential with second order poles at the source points. The quadratic differentials that arise in this way have a simple description in terms of their associated half-translation surfaces. Existence of stationary solutions is proved in some generality by solving an extremal problem involving an electrostatic energy functional, generalizing a classic problem studied by Teichm\"uller, Jenkins, Strebel and others. We study several special cases, including stationary Jordan curves on the Riemann sphere. We also introduce a discrete random version of the dynamics closely related to Propp's competitive erosion model, and conjecture that realizations of the lattice model will converge towards a corresponding solution to the competitive Hele-Shaw problem as the mesh size tends to zero.