INVARIANT PROBABILITY MEASURES FROM PSEUDOHOLOMORPHIC CURVES II: PSEUDOHOLOMORPHIC CURVE CONSTRUCTIONS

The paper proves that, under (A) Ψ_{W,M} ≠ 0 or (B) in dimension four with b2+(W)>1, c1(W)^2≠0, Ω|_{W−} exact, and c1(W−)≠0, every η-adapted cylinder (R×M,J,g) contains an ω-finite J-holomorphic curve (Theorems 1.3 and 1.4). Crucially, the authors avoid Symplectic Field Theory (SFT) neck-stretching because M need not be contact-type or stable Hamiltonian; they explicitly state that standard SFT neck-stretching is unusable in this generality and instead build a careful neck-stretching scheme plus trimming and apply Fish–Hofer’s exhaustive Gromov compactness to extract an ω-finite limit (see the discussion around Theorem 1.3/1.4, the neck-stretching construction, and the remarks on lack of λ-energy control: the outlines in Section 2 and Section 6.1–6.7, and the explicit warning that non-contact/non-stable M prevents SFT methods). The candidate solution, however, relies on SFT compactness and holomorphic buildings on a stable Hamiltonian hypersurface to pass to a symplectization level, assuming energy-splitting that the paper shows cannot be guaranteed outside the contact/stable case. This is a fundamental methodological error relative to the hypotheses of the paper (compare the paper’s statements on why SFT does not apply and the alternative Fish–Hofer route). Therefore, while the final conclusion matches the paper’s theorems, the model’s proof does not hold under the paper’s assumptions. See Theorem 1.3 and Theorem 1.4 with their hypotheses and goals, the neck-stretching set-up and diffeomorphism/tameness (and the lack of λ-energy control except near the neck boundary), and the exhaustive Gromov compactness strategy and ω-energy/genus bounds used to produce ω-finite curves (Theorems 1.3 and 1.4; Sections 2, 6.1–6.7; remarks on contact-type vs general case; and the existence statements from Gromov–Witten/Taubes used before stretching).