Proper pseudo-holomorphic maps between strictly pseudoconvex regions

Abstract

We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on an almost complex analogue of the scaling method. We also establish a link between the regularity of the extension and the regularity of the almost complex structures, and we determine explicit estimates for the H\"olderian norms. As a corollary, we get in the smooth case a necessary and sufficient condition on the almost complex structure of the target's space for the smooth extension of proper pseudo-holomorphic maps.