David Witt Nyström

The Hele-Shaw flow and moduli of holomorphic discs

We present a new connection between the Hele-Shaw flow, also known as two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this we prove short time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also starting from a smooth Jordan domain. Applying the same ideas we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

On the maximal rank problem for the complexhomogeneous Monge–Ampère equation

We give examples of regular boundary data for the Dirichlet problem for the complex homogeneous Monge-Ampere equation over the unit disc, whose solution is completely degenerate on a nonempty open set and thus fails to have maximal rank.

Canonical growth conditions associated to ample line bundles

We propose a new construction which associates to any ample (or big) line bundle L on a projective manifold X a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space TpX of any given point p. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows you to recover all the infinitesimal Okounkov bodies of L at p. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case the growth condition is ”equivalent” to the moment polytope. As in the toric case the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the wellknown connection between Seshadri constants and Gromov width established by McDuff and Polterovich.