Seminar

The unit 4-ball in R⁴ with the flat metric is a simple case of a hyperkähler 4-manifold with boundary; the metric on the 4-manifold has holonomy contained in SU(2)=Sp(1) and is, in particular, Ricci-flat and Kähler.  The boundary of a hyperkähler 4-manifold (for example, the unit 3-sphere) must necessarily have a certain structure on it, so a natural boundary value problem then arises: which of these structures on an oriented 3-manifold can arise as the boundary of a hyperkähler 4-manifold?  In particular, when can you deform the standard boundary structure on the 3-sphere and still “fill it in” to a hyperkähler metric on the 4-ball?  I will discuss these deformation problems for hyperkähler 4-manifolds with boundary, which is joint work with Joel Fine and Michael Singer.