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A. We give a construction of embedded contact homology (ECH) for a contact 3-manifold with convex sutured boundary and a pair of Legendrians Λ `and Λ ´contained in B satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of and Reeb chords of Λ `to Λ ´. The main ingredients include ‚ a general Legendrian adjunction formula for curves in R ˆ with boundary on R ˆΛ.‚ a relative writhe bound for curves in contact 3-manifolds asymptotic to Reeb chords.‚ a Legendrian ECH index and an accompanying ECH index inequality.The (action filtered) Legendrian ECH of any pair p , Λq of a closed contact 3-manifold and a Legendrian link Λ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer homology and ECH.The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.

A. Let Ă R 4 be a convex domain with smooth boundary . We use a relation between the extrinsic curvature of and the Ruelle invariant Rup q of the natural Reeb flow on to prove that there exist constants ą ą 0 independent of such that ă Rup q 2 volp q ¨sysp q ăHere sysp q is the systolic ratio of , i.e. the square of the minimal period of a closed Reeb orbit of divided by twice the volume of . We then construct dynamically convex contact forms on 3 that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact 3-spheres that are not strictly contactomorphic to a convex boundary .

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