An estimate on energy of min-max Seiberg-Witten Floer generators
Weifeng Sun
TL;DR
The article advances energy estimates for min-max Seiberg-Witten Floer generators to obtain a finite-index bound on the deviation between ECH-capacity-derived ratios and the ambient volume. By refining Taubes-type energy and action bounds and deriving differential and integral relations for key SW quantities, the authors show that the energy Ê(r) concentrates around r_k Vol(Y) with a quantified error, yielding the rate at which ECH capacities recover the volume. This directly strengthens the prior volume-recovery result and provides a quantitative speed of convergence, while also clarifying the existence and continuity properties of min-max generators. The work thus delivers both a sharper analytic bridge between ECH data and geometric volume and a constructive framework for min-max SW generators across perturbations.
Abstract
Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper "the asymptotics of ECH capacities" . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max Seiberg-Witten Floer generators. The second step used embedded balls in a certain symplectic four manifold. In this paper, stronger estimates on the energy of min-max Seiberg-Witten Floer generators are derived. This stronger estimate implies directly the "ECH capacities recover volume" theorem (without the help of embedded balls in a certain symplectic four manifold), and moreover, gives an estimate on its speed.