12-19【Richard Hind】Symplectic geometry of concave domains


报告人:Richard Hind(University of Notre Dame)



A concave domain is a subset $X$ of $\mathbb{C}^2$ such that $X = \pi^{-1} \Omega$, where $\pi : \mathbb{C}^2 \to \mathbb{R}^2_{\ge 0}, (z, w) \mapsto (|z|^2, |w|^2)$ is the moment map, and $\Omega$ is the region under the graph of a convex function $f$. We investigate these domains and symplectic embeddings between them. If $\Omega$ has finite area then $X$ is symplectomorphic to a bounded domain $Y$, although not necessarily one with smooth boundary, and we obtain lower bounds on the Minkowski dimension of $\partial Y$ in terms of the decay rate of $f$. More generally, this decay rate obstructs volume preserving symplectic embeddings and our $X$ provide the first examples of symplectic manifolds without packing stability. We also derive rough bounds on the Banach-Mazur distance, giving a general description of symplectic embeddings between concave domains. Our main tool will be the Embedded Contact Homology capacities, and in particular their subleading asymptotics. This is joint work with Dan Cristofaro-Gardiner.