Existence of a somewhere injective pseudo-holomorphic disc (Q1590001)

Let \((M,J)\) be a smooth almost complex manifold, \({\mathcal L}\) a totally real submanifold in \(M\), and \(u:\overline D\to M\) a smooth non-constant \(J\)-holomorphic disc with \(u(\partial D)\subset{\mathcal L}\). The author proves: There is another smooth \(J\)-holomorphic disc \(v\): \(\overline D\to M\) with \(v(D)\subset u(D)\), \(v(\partial D)\subset {\mathcal L}\), and an open dense subset \(\Omega\) of \(D\) with \(dv(z)\neq 0\) and \(v^{-1}v(z) =z\) for all \(z\in \Omega\), i.e. one gets a smooth simple \(J\)-holomorphic curve with boundary in \({\mathcal L}\) by restriction and factorization. The situation is illustrated by simple examples: the multi-covered disc, the lantern, the holed sphere. For higher genus, a similar result holds. In the case of discs, there is a consequence for symplectic manifolds: let \((M,\omega)\) be symplectic and \({\mathcal L}\) a strongly negative Lagrangian submanifold in \(M\), then \(M\) equipped with generic \((J,\omega)\) contains no non-constant \(J\)-holomorphic spheres and no non-constant \(J\)-holomorphic discs with boundary in \({\mathcal L}\).