Morse estimates for translated points on unit tangent bundles

TL;DR

This paper investigates the minimal number of translated points on the unit tangent bundle of Riemannian manifolds, proving existence results and Morse estimates for specific classes of manifolds, thus advancing understanding in contact topology.

Contribution

It establishes the existence of sequences of translated points with diverging time-shifts and provides Morse estimates for Zoll Riemannian manifolds, addressing conjectures in contact topology.

Findings

Existence of sequences of translated points with diverging time-shifts.

Morse estimates for the number of translated points on Zoll Riemannian manifolds.

Results support conjectures of Sandon in specific geometric contexts.

Abstract

In this article, we study conjectures of Sandon on the minimal number of translated points in the special case of the unit tangent bundle of a Riemannian manifold. We restrict ourselves to contactomorphisms of $SM$ that lift diffeomorphisms of $M$ homotopic to identity. We prove that there exist sequences $(p_n,t_n)$ where $p_n$ is a translated point of time-shift $t_n$ with $t_n\to+\infty$ for a large class of manifolds. We also prove Morse estimates on the number of translated points in the case of Zoll Riemannian manifolds.