Oberseminar Differentialgeometrie: Joel Fine (Brüssel), Vortrag: Knots, minimal surfaces and J-holomorphic curves
Monday, 18.01.2021 16:15 im Raum Zoom
Abstract: The asymptotic Plateau problem is as follows: given a submanifold K in the n-sphere, is there a minimal submanifold in (n+1)-dimensional hyperbolic space whose ideal boundary is K? I will explain how solving this problem when K is a knot (or link) in the 3-sphere leads to a knot invariant: the number of genus g minimal surfaces filling K depends on K only up to isotopy. One corollary is that any unknotted circle in the 3-sphere bounds a minimal disc in 4-dimensional hyperbolic space. In fact this count of minimal surfaces is actually an example of a Gromov-Witten type invariant: minimal surfaces in H^4 lift to J-holomorphic curves in the twistor space, which is a symplectic manifold. There is a class of infinite volume symplectic 6-manifolds, containing this twistor space, for which one can define counts of J-holomorphic curves which run out to infinity and this gives rise to more knot invariants in 3-manifolds. Finally, if there is time, I will speculate how to actually compute the invariant (and so relate it to known invariants). It appears that it should be possible to use simple combinatorics - a skein relation - to compute the invariant and so prove existence of minimal surfaces. Parts of this work are joint with Marcelo Alves.