Proof of the Arnold chord conjecture in three dimensions I

TL;DR

This paper proves that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord, using symplectic cobordisms and embedded contact homology, advancing the proof of the Arnold chord conjecture in three dimensions.

Contribution

It introduces a new approach linking symplectic cobordisms to embedded contact homology to prove the Arnold chord conjecture in three dimensions.

Findings

Every Legendrian knot has a Reeb chord in closed three-manifolds.

An exact symplectic cobordism induces a map on embedded contact homology.

The proof uses Seiberg-Witten theory in the sequel.

Abstract

This paper and its sequel prove that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord. The present paper deduces this result from another theorem, asserting that an exact symplectic cobordism between contact 3-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The latter theorem will be proved in the sequel using Seiberg-Witten theory.