  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.
