  This article is concerned with the question of whether an energy bound
implies a genus bound for pseudo-holomorphic curves in almost complex
manifolds. After reviewing what is known in dimensions other than 6, we
establish a new result in this direction in dimension 6; in particular, for
symplectic Calabi-Yau 6-manifolds. The proof relies on compactness and
regularity theorems for J-holomorphic currents.
