## 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

### Keywords:

Almost complex structures
Full Text:

### References:

 [1] Chirka, E. andRosay, J.-P., Remarks on the proof of a generalized Hartogs Lemma. Complex Analysis and Applications (Warsaw, 1997),Ann. Polon. Math. 70 (1998), 43–47. Correction inAnn. Polon. Math. 83 (2004), 289–290. [2] Ivashkovich, S. andRosay, J.-P., Schwarz-type Lemmas for solutions of $$\bar \partial$$ -equalities and complete hyperbolicity of almost complex manifolds,Ann. Inst. Fourier 54 (2004), 2387–2435. · Zbl 1072.32007 [3] Kruglikov, B. S., Existence of close pseudoholomorphic disks for almost complex manifolds and an application to the Kobayashi-Royden pseudonorm,Funktsional Anal. i Prilozhen. 33:1 (1999), 46–58, 96 (Russian). English transl.:Funct. Anal. Appl. 33 (1999), 38–48. · Zbl 0967.32024 [4] Lang, S.,Introduction to Complex Hyperbolic Spaces. Springer-Verlag, New York, 1987. · Zbl 0628.32001 [5] Nijenhuis, A. andWoolf, W. B., Some integration problems in almost-complex and complex manifolds,Ann. of Math. 77 (1963), 424–489. · Zbl 0115.16103 [6] Rosay, J.-P., A counterexample related to Hartog’s phenomenon (a question by E. Chirka),Michigan Math. J. 45 (1998), 529–535. · Zbl 0960.32020 [7] Royden, H. L., Remarks on the Kobayashi metric, inSeveral Complex Variables, II (College Park, MD, 1970), Lecture Notes in Math.185, pp. 125–137, Springer, Berlin, 1971. [8] Royden, H. L., The extension of regular holomorphic maps,Proc. Amer. Math. Soc. 43 (1974), 306–310. · Zbl 0292.32019 [9] Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, inHolomorphic Curves in Symplectic Geometry, Progress Math.117. pp. 165–189, Birkhäuser, Basel. 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.