Knots, minimal surfaces and J-holomorphic curves

Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space, H4. The main theme of my talk is that it should be possible to count minimal surfaces in H4 which fill K and obtain a link invariant. In other words, the count doesn’t change under isotopies of K. When one counts minimal disks, this is a theorem. Unfortunately there is currently a gap in the proof for more complicated surfaces. I will explain “morally” why the result should be true and how I intend to fill the gap. In fact, this (currently conjectural) invariant is a kind of Gromov–Witten invariant, counting J-holomorphic curves in a certain symplectic 6-manifold diffeomorphic to S2×H4. The symplectic structure becomes singular at infinity, in directions transverse to the S2 fibres. These singularities mean that both the Fredholm and compactness theories have fundamentally new features, which I will describe. Finally, there is a whole class of infinite-volume symplectic 6-manifolds which have singularities modelled on the above situation. I will explain how it should be possible to count J-holomorphic curves in these manifolds too, and obtain invariants for links in other 3-manifolds.