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Upper semi-continuity of the Kobayashi-Royden pseudo-norm, a counterexample for Hölderian almost complex structures. (English) Zbl 1091.32009

Let \(X\) be an almost complex manifold, with an almost complex structure tensor \(J\) of class \(C^\alpha\), for some \(\alpha >0\). Then, by a result of Nijenhuis-Woolf it is possible to construct for any \((p,V)\in TX \) (the tangent bundle to \(X\)) a \(J\)-holomorphic disc through \(p\) with tangent vector \(V\) at the point \(p\). This allows to define the Kobayashi-Royden pseudonorm \(F_X(p,V)\) on \(TX\). It is known that \(F_X\) is upper semicontinuous, provided that \(J\) is \(C^{1,\alpha}\) (Ivashkovich-Rosay).
In the article under review the authors show, that the condition of Hölder continuity of \(J\) is in general too weak, as that it could imply upper semicontinuity of \(F_X\). They show
Theorem: On the unit bidisc \(X\) in \(\mathbb{C}^2\) there exists an almost complex structure \(J\) of class \(C^{1/2}\) such that \(F_X\) is not upper semicontinuous.

MSC:

32Q60 Almost complex manifolds
32F45 Invariant metrics and pseudodistances in several complex variables
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