The monopole equations and \(J\)-holomorphic curves on weakly convex almost Kähler 4-manifolds (Q2706610)

The main result of the paper is an extension of results of \textit{C. H. Taubes} [J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)] to weakly convex almost Kähler manifolds, which are non-compact in general. It is proved that a weakly convex almost Kähler 4-manifold contains a compact, non-constant \(J\)-holomorphic curve if the corresponding monopole invariant, which is equivalent to its Gromov-Witten invariant, is non-zero and if the corresponding line bundle is non-trivial. An application of this result to contact topology is given. It is proved that if \((X_0,\omega)\) is a minimal symplectic filling of the contact space \(\text{SU}(2)/\Gamma\), then the intersection form of \(X_0\) is negative definite, and the trivialization of the canonical line bundle \({\mathcal K}_{X_0}\) over the boundary \(\partial X_0\) extends to the interior of \(X_0\), where \(X_0\) must be a spin manifold.