An estimate on energy of min-max Seiberg-Witten Floer generators

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.