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# Mathematical Research Letters

## Volume 26 (2019)

### Number 6

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

Pages: 1807 – 1827

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n6.a10

#### Author

#### Abstract

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 $k \to + \infty$ 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 $k \to + \infty$ 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 $k \to \infty$ 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 $k \to \infty$) 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.

Received 4 February 2018

Accepted 22 July 2019

Published 6 March 2020