Contact instantons, anti-contact involution and proof of Shelukhin's conjecture

TL;DR

This paper proves Shelukhin's conjecture on translated points in contact geometry using contact instantons and involutive symmetries, advancing understanding of contactomorphisms and their fixed points.

Contribution

It introduces a novel approach combining contact instantons with involutive symmetries to prove a longstanding conjecture in contact topology.

Findings

Proof of Shelukhin's conjecture for all closed contact manifolds.

Development of contact Hamiltonian geometry with involutive symmetry.

Application of bordered contact instantons with Legendrian boundary conditions.

Abstract

In this paper, we prove Shelukhin's conjecture on the translated points on any closed contact manifold $(Q,\xi)$ which reads that for any choice of function $H = H(t,x)$ and contact form $\lambda$ the contactomorphism $\psi_H^1$ carries a translated point in the sense of Sandon, whenever the inequality $$ \|H\| \leq T(\lambda,M) $$ holds the case. Main geometro-analytical tools are those of bordered contact instantons employed in [Ohc] with Legendrian boundary condition via the Legendrianization of contact diffeomorphisms. Along the way, we utilize the functorial construction of the contact product that carries an involutive symmetry and develop relevant contact Hamiltonian geometry with involutive symmetry. This involutive symmetry plays a fundamental role in our proof in combination with the analysis of contact instantons.