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

In [1], Cristofaro-Gardiner, Hutchings and Ramos proved that embedded contact homology (ECH) capacities of a 4-dimensional symplectic manifold can recover the volume. In particular, a certain sequence of ratios constructed from ECH capacities, indexed by positive integers, was shown to converge to the volume in the index k → +∞ limit. There were two main steps in [1] to proving this theorem: The first step used estimates for the energy of min-max Seiberg-Witten Floer generators to see that the k → +∞ limit of the ratios was a lower bound for the volume. The second step used embedded balls in a certain symplectic four manifold to prove that the k →∞ limit of the ratios was an upper bound. Stronger estimates on the energy of min-max Seiberg-Witten Floer generators are derived in this paper that give an effective bound for finite index k on the norm of the difference between the ECH ratio at index k and the volume. This bound implies directly (by taking k →∞ ) the theorem in [1] that ECH capacities recover volume. Section 1 introduces the prior knowledge and the main theorem. Section 2 and Section 3 prove the main theorem. Section 4 is an addendum that talks about the Seiberg-Witten Floer min-max generators.