Symplectic embeddings into disk cotangent bundles

In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles D*S-2 and D*RP2. 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*RP2 using a direct limit argument on the action inspired by Bourgeois Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable system 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.