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

Yong-Geun Oh

TL;DR

This work proves Shelukhin's conjecture on translated points for any compact contact manifold by coupling bordered contact instanton analysis with a ${\mathbb Z}_2$-equivariant framework built from a contact product $M_Q=Q\times Q\times {\mathbb R}$ and its anti-contact involution. Translated points for a contactomorphism generated by $H$ are translated into Reeb chords between Legendrian graphs via Legendrianization, while a taming structure and domain-dependent lifted CR geometry ensure robust $C^0$ and energy estimates. The approach yields existence results under the chord period bound ${\|H\|} \le T(M,\lambda)$ and, in the nondegenerate strict case, a lower bound matching ${\dim} H^*(M;{\mathbb Z}_2)$, with Cant–Hedicke showing sharpness. The method also clarifies the relation to other frameworks (e.g., Rabinowitz Floer homology, Chekanov–Eliashberg DGA) and introduces the untangling energy perspective $e^{\text{trn}}_\lambda$, highlighting the quantitative impact on contact dynamics.

Abstract

In this paper, we prove Shelukhin's conjecture on the translated points on any closed contact manifold $(Q,ξ)$ which reads that for any choice of function $H = H(t,x)$ and contact form $λ$ the contactomorphism $ψ_H^1$ carries a translated point in the sense of Sandon, whenever the inequality $$ \|H\| \leq T(λ,M) $$ 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.

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

TL;DR

This work proves Shelukhin's conjecture on translated points for any compact contact manifold by coupling bordered contact instanton analysis with a -equivariant framework built from a contact product and its anti-contact involution. Translated points for a contactomorphism generated by are translated into Reeb chords between Legendrian graphs via Legendrianization, while a taming structure and domain-dependent lifted CR geometry ensure robust and energy estimates. The approach yields existence results under the chord period bound and, in the nondegenerate strict case, a lower bound matching , with Cant–Hedicke showing sharpness. The method also clarifies the relation to other frameworks (e.g., Rabinowitz Floer homology, Chekanov–Eliashberg DGA) and introduces the untangling energy perspective , highlighting the quantitative impact on contact dynamics.

Abstract

In this paper, we prove Shelukhin's conjecture on the translated points on any closed contact manifold which reads that for any choice of function and contact form the contactomorphism carries a translated point in the sense of Sandon, whenever the inequality 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.
Paper Structure (33 sections, 45 theorems, 248 equations)

This paper contains 33 sections, 45 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Statement of main result
  3. Contact diffeomorphisms and their Legendrianizations
  4. Contact product $M_Q$ and anti-contact involution
  5. Contact-product and perturbed contact instantons
  6. Discussion and the relationship with other approaches
  7. Basics of contact Hamiltonian calculus
  8. Contact isotopy, conformal exponent and contact Hamiltonian
  9. Developing Hamiltonians and their calculus
  10. Perturbed contact instantons and their subsequence convergence
  11. Contact product and its Hamiltonian geometry
  12. Contact product and Legendrianization
  13. Definition of contact product
  14. Contact product $Q \star Q$ and anti-contact involution
  15. Contact product and ${\mathbb Z}_2$-symmetry
  16. ...and 18 more sections

Key Result

Theorem 1.3

Assume $(M,\xi)$ is a compact contact manifold and let $\psi \in \operatorname{Cont}_0(M,\xi)$. Let $\lambda$ be a contact form of $\xi$ and denote by $T(M,\lambda) > 0$ the minimal period of closed Reeb orbits of $\lambda$. Then for any $H \mapsto \psi$, the following holds:

Theorems & Definitions (88)

  • Definition 1.1: Sandon sandon:translated
  • Definition 1.2
  • Theorem 1.3: Main Theorem; Shelukhin's conjecture
  • Remark 1.4
  • Definition 1.5: Definition 24 shelukhin:contactomorphism
  • Definition 1.6: Definition 1.24 oh:entanglement1
  • Definition 1.7: Untanglement energy of $\lambda$
  • Corollary 1.8
  • Remark 1.9
  • Definition 1.10: $\Gamma_{\psi}$
  • ...and 78 more