2015/09/09 by Ross, Julius, Nystrom, David Witt
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1509.02665
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 P1× \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 ℙ1× ∂ \mathbbD. Second, we produce explicit families of smooth geodesic rays in the space of Kähler metrics on \mathbb P1 and on the unit disc \mathbb D that are constructed from an exhausting family of increasing smoothly varying simply connected domains.