Packing Densities of Delzant and Semitoric Polygons

Yu Du   ; Gabriel Kosmacher   ; Yichen Liu   ; Jeff Massman   ; Joseph Palmer   ; Timothy Thieme   ; Jerry Wu   ; Zheyu Zhang

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

Exploiting the relationship between $4$-dimensional toric and semitoric integrable systems with Delzant and semitoric polygons, respectively, we develop techniques to compute certain equivariant packing densities and equivariant capacities of these systems by working exclusively with the polygons. This expands on results of Pelayo and Pelayo–Schmidt. We compute the densities of several important examples and we also use our techniques to solve the equivariant semitoric perfect packing problem, i.e., we list all semitoric polygons for which the associated semitoric system admits an equivariant packing which fills all but a set of measure zero of the manifold. This paper also serves as a concise and accessible introduction to Delzant and semitoric polygons in dimension four.

Détail

Keywords: equivariant packing, equivariant symplectic capacities, semitoric integrable systems, semitoric polygons, integrable systems.

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