Geometric Analysis and Topology Seminar

---

Knots, minimal surfaces and J-holomorphic curves

Speaker: Joel Fine, Universite Libre de Bruxelles

Location: Online

Videoconference link: https://nyu.zoom.us/j/94747124588

Date: Wednesday, March 23, 2022, 11 a.m.

Synopsis:

Let K be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to K, and in this way obtain a knot invariant. In other words the number of minimal discs depends only on the isotopy class of the boundary. “Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how these counts of minimal discs can be seen as Gromov-Witten invariants for the twistor space of hyperbolic 4-space. Whilst Gromov-Witten theory suggests the overall strategy for defining the minimal surface link-invariant, there are significant differences in how to actually implement it. This is because the geometry of both hyperbolic space and its twistor space become singular at infinity. As a consequence, the PDEs involved (both the minimal surface equation and J-holomorphic curve equation) are degenerate rather than elliptic at the boundary. I will try and explain how to overcome these complications.