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

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 $$\mathrm{\parallel }H\mathrm{\parallel }\le T(\lambda ,M)$$ 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.