Symplectic embeddings into disk cotangent bundles

TL;DR

This paper computes the embedded contact homology capacities of disk cotangent bundles of spheres and projective planes, and finds explicit sharp symplectic embeddings into these domains using advanced techniques.

Contribution

It explicitly calculates ECH capacities for disk cotangent bundles and constructs sharp symplectic embeddings using integrable systems techniques.

Findings

Computed ECH capacities of D^*S^2 and D^*RP^2

Established symplectomorphisms of certain cotangent bundles to standard domains

Determined Gromov widths of these cotangent bundles

Abstract

In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles $D^*S^2$ and $D^*\mathbb{R} P^2$. We also find sharp symplectic embeddings into these domains. In particular, we compute their Gromov widths. In order to do that, we explicitly calculate the ECH chain complexes of $S^*S^2$ and $S^* \mathbb{R} P^2$ using a direct limit argument on the action inspired by Bourgeois's Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable systems techniques to find explicit symplectic embeddings. In particular, we prove that the disk cotangent bundles of a hemisphere and of a punctured sphere are symplectomorphic to an open ball and a symplectic bidisk, respectively.