Seiberg–Witten solutions converge to gradient-flow t-orbits

Let Y+Y^+ be a cylindrical-end 33-manifold with harmonic Morse function f+f^+, let ZZ be the set of its index-11 and index-22 critical points, and let a t-orbit be a weighted collection of 11-submanifolds tangent to f+-\nabla f^+ with the stated linking-number conditions. Let Γ=r1,r2,\Gamma=\\{r_1,r_2,\ldots\\} be an unbounded sequence in (r0,)(r_0,\infty), let wrw_r be closed 22-forms satisfying the perturbation constraints and wrC5ζ\lVert w_r\rVert_{C^5}\leq\zeta, and let crZr,wrSW(Y+,s,f+)\mathfrak{c}_r\in\mathcal{Z}^{\rm SW}_{r,w_r}(Y^+,\mathfrak{s},f^+) be corresponding solutions. Seiberg–Witten-to-t-orbit convergence conjecture. The sequence crr\\{\mathfrak{c}_r\\}_r t-converges to a t-orbit. Four-dimensional analogues are cited, but the three-dimensional assertion itself is not resolved in the source.

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Primary source

Yi-Jen Lee, “Morse theory and Seiberg-Witten moduli spaces of 3-dimensional cobordisms, I”, arXiv:2412.20710 (2025).

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