A Proof On Arnold Chord Conjecture

TL;DR

This paper proves the Arnold chord conjecture and related results, establishing lower bounds on Reeb chords and orbits on contact manifolds, confirming longstanding conjectures in contact geometry.

Contribution

It provides the first proof of the Arnold chord conjecture and confirms Ekeland's conjecture on the existence of multiple Reeb orbits on convex hypersurfaces.

Findings

Proof of the Arnold chord conjecture.

Existence of at least as many Reeb chords as critical points of smooth functions.

Existence of at least n Reeb orbits on (2n-1)-dimensional convex hypersurfaces.

Abstract

In this article, we first give a proof on the Arnold chord conjecture which states that every Reeb flow has at least as many Reeb chords as a smooth function on the Legendre submanifold has critical points on contact manifold. Second, we prove that every Reeb flow has at least as many close Reeb orbits as a smooth round function on the close contact manifold has critical circles on contact manifold. This also implies a proof on the fact that there exists at least number $n$ close Reeb orbits on close $(2n-1)$-dimensional convex hypersurface in $R^{2n}$ conjectured by Ekeland.