Knots, minimal surfaces and J-holomorphic curves

Abstract: Let K be a knot in the 3-sphere. I will explain how one can count minimal discs 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 knot. I think it should actually be possible to define a family of link invariants, counting minimal surfaces filling links, but at this stage this is still just a conjecture. “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.

algebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

Geometria em Lisboa (IST)

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