Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$

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.