Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture

TL;DR

This paper proves Weinstein's and Arnold's conjectures using contact instanton theory, establishing fundamental classes and transversality results that advance the understanding of contact Hamiltonian dynamics.

Contribution

It provides the first full proofs of Weinstein's and Arnold's conjectures in contact geometry leveraging contact instanton moduli spaces and new transversality techniques.

Findings

Proof of Weinstein's conjecture in full generality.

Proof of Arnold's chord conjecture in full generality.

Development of transversality methods for contact instantons.

Abstract

The present paper is a continuation of the study of the interplay between the contact Hamiltonian dynamics and the moduli theory of (perturbed) contact instantons and its applications initiated in [Oh21b, Oh22a]. In this paper we prove Weinstein's conjecture and Arnold's chord conjecture in their full generalities. The two key ingredients lying in the background of our proof of Arnold's chord conjecture are the existence of the fundamental class of the Legendrian contact instanton cohomology modulo bubbling-off, and the evaluation transversality of the moduli space of contact instantons against the level set of conformal exponent function. Our proof of Weinstein's conjecture also utilizes the existence scheme of translated points of a contactomorphism developed in [Oh22a], especially associated to a contact Hamiltonian loop, via the geometric construction of the Legendrianization of contactomorphisms of $(Q,\lambda)$ in the contact product $M_Q = Q \times Q \times \mathbb R$ and its usage of the $\mathbb Z_2$ anti-contact involutive symmetry.