Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$
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| Format: | Preprint |
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2025
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| _version_ | 1866918156347375616 |
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| author | Fernandes, Rafael Ferreira, Brayan |
| author_facet | Fernandes, Rafael Ferreira, Brayan |
| contents | In this paper, we make use of elementary spectral invariants given by the max-min energy of pseudoholomorphic curves, recently defined by Michael Hutchings, to study periodic $3$-dimensional Reeb flows. We prove that Zoll contact forms on $S^3$ are characterized by $c_1 = c_2 = \mathcal{A}_{\min}$. This follows from the spectral gap closing bound property and a computation of ECH spectral invariants for Zoll contact forms defined on Lens spaces $L(p,1)$ for $p\geq 1$. The former characterization fails for Lens spaces $L(p,1)$ with $p>1$. Nevertheless, we characterize Zoll contact forms on $L(p,1)$ in terms of ECH spectral invariants. Lastly, we note a characterization of Besse contact forms also holds for elementary spectral invariants analogously to the one obtained by Dan Cristofaro-Gardiner and Mazzucchelli. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$ Fernandes, Rafael Ferreira, Brayan Symplectic Geometry Dynamical Systems 53Dxx, 53E50, 34C25, 37C55 In this paper, we make use of elementary spectral invariants given by the max-min energy of pseudoholomorphic curves, recently defined by Michael Hutchings, to study periodic $3$-dimensional Reeb flows. We prove that Zoll contact forms on $S^3$ are characterized by $c_1 = c_2 = \mathcal{A}_{\min}$. This follows from the spectral gap closing bound property and a computation of ECH spectral invariants for Zoll contact forms defined on Lens spaces $L(p,1)$ for $p\geq 1$. The former characterization fails for Lens spaces $L(p,1)$ with $p>1$. Nevertheless, we characterize Zoll contact forms on $L(p,1)$ in terms of ECH spectral invariants. Lastly, we note a characterization of Besse contact forms also holds for elementary spectral invariants analogously to the one obtained by Dan Cristofaro-Gardiner and Mazzucchelli. |
| title | Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$ |
| topic | Symplectic Geometry Dynamical Systems 53Dxx, 53E50, 34C25, 37C55 |
| url | https://arxiv.org/abs/2510.06496 |