Regularity and estimates for J -holomorphic discs attached to a maximal totally real submanifold

We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity to the boundary from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of Hölderian norms of such discs. © 2007 Elsevier Inc. All rights reserved