Applications of the duality between the Complex Monge-Ampère Equation and the Hele-Shaw flow

Abstract

We give two applications of the the duality between the complex Homogeneous Monge-Ampère Equation (HMAE) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HMAE over ${\mathbb{P}}^1\times \overline{\mathbb{D}}$ 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 ${\mathbb{P}}^1\times \partial \mathbb{D}$. Second, we produce explicit families of smooth geodesic rays in the space of Kähler metrics on ${\mathbb{P}}^1$ and on the unit disc $\mathbb{D}$ that are constructed from an exhausting family of increasing smoothly varying simply connected domains.

2 pages, 1 figure. Minor corrections and exposition changes following referee comments

Cited by in corpus (1)

• On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampère Equation