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 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.
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Notes
It is a simple calculation to verify that (6) has exactly two solutions when \(h_{\mathrm{min}}^{\varepsilon ,C}(j)<h\) and one solution when \(h_{\mathrm{min}}^{\varepsilon ,C}=h\).
We recall that since \(\gamma _{p_1}^i\) is nullhomologous, i is a multiple of 4.
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Acknowledgements
We would like to thank Joé Brendel, Jo Nelson, Felix Schlenk and Egor Shelukhin for helpful conversations. The second author is partially supported by a grant from the Serrapilheira Institute, the FAPERJ grant Jovem Cientista do Nosso Estado and the CNPq grants 407510/2018-4 and 306405/2020-2.
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