Symplectic embeddings into disk cotangent bundles

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