Joel Fine

I will describe work in progress, parts of which are joint with Marcelo Alves. Let M be a compact 4-manifold with boundary Y, and let g be a negatively curved Riemannian metric on the interior of M, which is asymptotically hyperbolic and which makes Y totally umbilic. (This conformally invariant condition makes sense even though the metric is not defined at Y.) Given a knot or link K in Y, I will explain how one can count the minimal surfaces in M which have K as their ideal boundary. This gives a knot invariant - the answer depends only on K up to isotopy. Moreover, it appears that the answer depends only on the metric g up to isotopy also. I will explain the ideas in the proofs as well as some natural questions that these ideas lead to.