Closed Reeb orbits on contact type hypersurfaces in $T^*S^n$

TL;DR

This paper proves the existence of multiple closed Reeb orbits on certain hypersurfaces in cotangent bundles of spheres, under specific convexity and non-degeneracy conditions, advancing understanding in contact geometry.

Contribution

It establishes lower bounds on the number of closed Reeb orbits under dynamical convexity and non-degeneracy assumptions in cotangent bundles.

Findings

At least [ (n+1)/2 ] closed Reeb orbits exist under dynamical convexity.

Presence of at least two irrationally elliptic closed Reeb orbits when the contact form is non-degenerate.

Results apply to hypersurfaces enclosing the zero section in T* S^n.

Abstract

In this paper, it is proved that under dynamically convex condition, there exist at least $[\frac{n+1}{2}]$ closed Reeb orbits on a closed contact type hypersurface in $T^*S^n$ enclosing the zero section and bounding a simply connected Liouville domain. Furthermore, if the contact form is non-degenerate and has finitely many closed Reeb orbits, then there exist at least two irrationally elliptic closed Reeb orbits.