An elementary alternative to PFH spectral invariants

Abstract

Inspired by Hutchings' elementary alternative to ECH capacities, we introduce an elementary alternative to spectral invariants defined via periodic Floer homology (PFH). We use these spectral invariants to provide more elementary proofs of a number of results which have recently been obtained using PFH spectral invariants. Among these results are quantitative closing lemmas for area-preserving surface diffeomorphisms and the simplicity conjecture. 2.1. Almost complex structures 10 2.2. Pseudo-holomorphic curves and energy 11 2.3. Definition of c d,k 12 2.4. Neck stretching 13 2.5. Proofs of basic properties 15 3. Applications 18 3.1. Quantitative closing lemmas 18 3.2. Simplicity conjecture and extension of Calabi 19 Appendix A. Properties of the relative action spectrum 25 Appendix B. Computation of Gromov-Taubes invariants 28 B.1. Review of Gromov-Taubes invariants 28 B.2. Computations without Seiberg-Witten theory 28 B.3. Computations with Seiberg-Witten theory 31 References 32