We give two applications of the duality between the Homogeneous Complex Monge-Ampere Equation (HCMA) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HCMA over P-1 x (D) over bar is not twice differentiable at a given collection of points, and also examples that are not twice differentiable along a set of codimension one in P-1 x partial derivative D we discuss how to obtain explicit families of smooth geodesic rays in the space of Kahler metrics on P-1 and on the unit disc D that are constructed from an exhausting family of increasing smoothly varying simply connected domains.
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