Estimates of the Kobayashi metric on almost complex manifolds

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We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold (M,J) admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in M and to give a sufficient condition for the complete hyperbolicity of a domain in (M,J). Finally we obtain the regularity up to the bounday of J-holomorphic discs attached to a totally real submanifold in M.