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Generic jet evaluation transversality of contact instantons against contact distribution

Yomg-Geun Oh

TL;DR

The authors establish generic $0$-jet interior and boundary evaluation transversality and $1$-jet evaluation transversality for moduli spaces of contact instantons on a contact manifold $(M,\\Xi)$, by developing a robust off-shell Fredholm framework and employing Sard–Smale techniques. Central to the approach is the universal moduli space of solutions and its linearization, with careful analysis of how evaluation maps interact with the contact distribution via the operators $\\overline{\\partial}^\\pi$ and $d(w^*\\lambda\circ j)$. The work provides rigorous transversality results against $\\Xi$ for both $0$- and $1$-jet constraints, including a detailed treatment of interior and boundary marked points, Reeb chords, and Legendrian boundary data. These transversality results underpin key steps in Weinstein-type questions and the construction of contact-invariant counts, showing that for a residual set of perturbed data, the relevant moduli spaces are smooth and of expected dimension. The combination of off-shell analysis, adjoint arguments, and unique continuation yields a solid foundation for the higher-jet transversality program in contact and Legendrian settings, with potential applications to Weinstein conjectures and Ramified Gromov–Witten-type theories in contact geometry.

Abstract

For a given coorientable contact manifold $(M,Ξ)$ with contact distribution $Ξ$, we consider its contact forms $λ$ with $\ker λ= Ξ$, and the associated contact triads $(M,λ, J)$. For a generic choice of contact form $λ$, we prove the (0-jet) the interior and boundary evaluation maps, and the 1-jet transversality of contact instantons (against contact distribution, for example).

Generic jet evaluation transversality of contact instantons against contact distribution

TL;DR

The authors establish generic -jet interior and boundary evaluation transversality and -jet evaluation transversality for moduli spaces of contact instantons on a contact manifold , by developing a robust off-shell Fredholm framework and employing Sard–Smale techniques. Central to the approach is the universal moduli space of solutions and its linearization, with careful analysis of how evaluation maps interact with the contact distribution via the operators and . The work provides rigorous transversality results against for both - and -jet constraints, including a detailed treatment of interior and boundary marked points, Reeb chords, and Legendrian boundary data. These transversality results underpin key steps in Weinstein-type questions and the construction of contact-invariant counts, showing that for a residual set of perturbed data, the relevant moduli spaces are smooth and of expected dimension. The combination of off-shell analysis, adjoint arguments, and unique continuation yields a solid foundation for the higher-jet transversality program in contact and Legendrian settings, with potential applications to Weinstein conjectures and Ramified Gromov–Witten-type theories in contact geometry.

Abstract

For a given coorientable contact manifold with contact distribution , we consider its contact forms with , and the associated contact triads . For a generic choice of contact form , we prove the (0-jet) the interior and boundary evaluation maps, and the 1-jet transversality of contact instantons (against contact distribution, for example).
Paper Structure (16 sections, 32 theorems, 249 equations)

This paper contains 16 sections, 32 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Off-shell Fredholm setting for the evaluation transversality
  4. Off-shell description of moduli spaces
  5. Inclusion of marked points and evaluation maps
  6. Fredholm theory of the linearized operator
  7. Fredholm theory on punctured bordered Riemann surfaces
  8. Generic mapping transversality of (relative) contact instantons
  9. The (0-jet) evaluation transversality: Statement
  10. The (0-jet) evaluation transversality: Proof
  11. Generic 1-jet evaluation transversality: Statement
  12. Generic 1-jet evaluation transversality: Proof
  13. Geometric description of the off-shell framework
  14. Evaluation transversality and its off-shell function spaces
  15. Proof of the evaluation transversality against $\Xi$
  16. ...and 1 more sections

Key Result

Lemma 1.1

Let $\lambda$ be a contact form of a contact manifold $(M,\Xi)$. Consider $\varphi: N \to M$ be any smooth map. Then the differential is transversal to $\Xi$ if and only if the function $TN \to {\mathbb R}$ defined by $(x,v) \mapsto \lambda_x(d\varphi(v))$ has 0 as a regular value.

Theorems & Definitions (51)

  • Lemma 1.1
  • Definition 1.2: Moduli space with $\Xi$-tangency condition
  • Theorem 1.3: $1$-jet evaluation transversality
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Iso-speed Reeb chords oh:entanglement1
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • ...and 41 more