Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures

Abstract

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class $\CC^\alpha$, for some $\alpha >0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class $\CC^{1,\alpha}$ in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden's proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that $\CC^{1,\alpha}$ regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class $\CC^{1\over 2}$ on the unit bidisc $\D^2\subset \C^2$, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.