×

zbMATH — the first resource for mathematics

On the laminarity of certain currents. (Sur la laminarité de certains courants.) (French) Zbl 1061.32005
The aim of this paper is the following theorem: Let \(C_n\) be a sequence of smooth analytic curves of the unit ball \(\mathbb B\subset\mathbb C^2\) (spreading a little beyond \(\overline{\mathbb B}\)). Let \(A_n\) be the area of \(C_n\), and \(G_n\) be the genus of \(C_n\). Suppose that the sequence of currents \(T_n=[C_n]/A_n\) converges toward a (1,1)-positive closed current \(T\) of \(\mathbb B\). If \(G_n = O(A_n)\), then \(T\) is laminar.

MSC:
32C30 Integration on analytic sets and spaces, currents
32C25 Analytic subsets and submanifolds
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Bedford E. , Lyubich M. , Smillie J. , Polynomial diffeomorphisms of C 2 . IV : The measure of maximal entropy and laminar currents , Invent. Math. 112 ( 1993 ) 77 - 125 . MR 1207478 | Zbl 0792.58034 · Zbl 0792.58034 · doi:10.1007/BF01232426 · eudml:144096
[2] Bedford E. , Smillie J. , Polynomial diffeomorphisms of C 2 . V : Critical points and Lyapunov exponents , J. Geom. Anal. 8 ( 1998 ) 349 - 383 . MR 1707733 | Zbl 1004.37027 · Zbl 1004.37027 · doi:10.1007/BF02921791
[3] Bishop E. , Conditions for the analyticity of certain sets , Michigan Math. J. 11 ( 1964 ) 289 - 304 . Article | MR 168801 | Zbl 0143.30302 · Zbl 0143.30302 · doi:10.1307/mmj/1028999180 · minidml.mathdoc.fr
[4] Cantat S. , Dynamique des automorphismes des surfaces K3 , Acta Math. 187 ( 2001 ) 1 - 57 . MR 1864630 | Zbl 1045.37007 · Zbl 1045.37007 · doi:10.1007/BF02392831
[5] Dujardin R. , Laminar currents in P 2 , à paraître dans Math. Ann. Zbl 1021.37018 · Zbl 1021.37018 · doi:10.1007/s00208-002-0402-9
[6] Duval J. , Sibony N. , Polynomial convexity, rational convexity, and currents , Duke Math. J. 79 ( 1995 ) 487 - 513 . Article | MR 1344768 | Zbl 0838.32006 · Zbl 0838.32006 · doi:10.1215/S0012-7094-95-07912-5 · minidml.mathdoc.fr
[7] Lelong P. , Propriétés métriques des variétés analytiques complexes définies par une équation , Ann. Sci. Éc. Norm. Sup. 67 ( 1950 ) 393 - 419 . Numdam | MR 47789 | Zbl 0039.08804 · Zbl 0039.08804 · numdam:ASENS_1950_3_67__393_0 · eudml:81651
[8] Morgan F. , Geometric Measure Theory , Academic Press , 1988 . MR 933756 | Zbl 0671.49043 · Zbl 0671.49043
[9] Nevanlinna R. , Analytic Functions , Springer-Verlag , 1970 . MR 279280 | Zbl 0199.12501 · Zbl 0199.12501
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.