Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures

Details Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures

2004-02-20

arxiv.org/abs/math/0402331v1

Mathematics

Complex Variables

5 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-709293