Boundary-value problems for 4-dimensional HyperKaehler metrics

Michael Singer (University College London)

Abstract: In four dimensions, hyperKaehler structures are essentially the
same as triples of closed 2-forms which satisfy a suitable orthonormality
condition.  This description in terms of triples suggests a natural
boundary-value problem for 4-dimensional hyperKaehler structures in terms of
framings of the boundary N.  (This was known to Robert Bryant, and,
possibly, to Elie Cartan.)   The question of which boundary framings of N
can be filled by hyperKaehler triples seems very difficult, but I shall
describe the perturbative version of this problem.  This talk is based on
joint work with Joel Fine and Jason Lotay.