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- W1506817062 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 $pin X$ 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 $CC^{1over 2}$ on the unit bidisc $D^2subset C^2$, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle." @default.
- W1506817062 created "2016-06-24" @default.
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- W1506817062 date "2005-10-01" @default.
- W1506817062 modified "2023-09-24" @default.
- W1506817062 title "Upper semi-continuity of the Kobayashi-Royden pseudo-norm, a counterexample for Hölderian almost complex structures" @default.
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- W1506817062 doi "https://doi.org/10.1007/bf02384787" @default.
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