Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$
Rafael Fernandes, Brayan Ferreira
TL;DR
The paper develops a framework that uses elementary max-min energy invariants $c_k(Y,\lambda)$, defined via pseudoholomorphic curves, to detect Zoll and Besse Reeb flows on 3-manifolds. By connecting these invariants to the embedded contact homology (ECH) spectrum and its $U$-map, the authors prove that Zoll on $S^3$ is equivalent to $c_1=c_2=\mathcal{A}_{min}$, and they extend a similar criterion to Zoll structures on lens spaces $L(p,1)$ through $c^{ECH}$-invariants. A parallel Besse criterion is established via $c_k(Y,\lambda)=c_{k+1}(Y,\lambda)$ for some $k$, mirroring prior ECH results. The analysis combines meromorphic-section techniques for Zoll fibrations and explicit ECH computations on Zoll lens spaces to reveal when the max-min invariants capture Zoll data and when ECH invariants provide finer dynamical information. Collectively, the work links low-level symplectic invariants to global dynamical structure of Reeb flows, yielding computable criteria and clarifying the differences between $S^3$ and lens-space settings.
Abstract
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.