GT seminar: "Spectral diameter and Lagrangian capacity"

Abstract: I will present some aspects of two papers (arXiv:2408.07214 and arXiv:2409.14142) concerning spectral invariants for compactly supported Hamiltonian systems in Cn. First, I will explain our discovery in arXiv:2408.07214 that the spectral diameter of a ball B(a) (of capacity a) is half of that of a cylinder Z(a). Thus the spectral diameter is not a normalized capacity. Then I will explain how this fact is used to prove certain cases of the conjecture that: any closed Lagrangian L in a ball B(1) in Cn, n>1, bounds holomorphic disks with area at most 1/2 (note that the Lagrangian lift of RPn-1 inside the boundary of B(1) bounds disks with area 1/2, so the conjecture cannot be improved). Our proof is based on a delicate version of the Lagrangian control property which incorporates spectral invariants, boundary depth, and the minimal area of holomorphic disks.

This is a talk in our Geometry and Topology Seminar Series.