Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T07:04:46.075Z Has data issue: false hasContentIssue false
  • English
  • Français

PLURIPOLAR SETS, REAL SUBMANIFOLDS AND PSEUDOHOLOMORPHIC DISCS

Part of: Global differential geometry Holomorphic mappings and correspondences

Published online by Cambridge University Press:  08 April 2019

ALEXANDRE SUKHOV
ALEXANDRE SUKHOV*
Affiliation:
Université des Sciences et Technologies de Lille, Laboratoire Paul Painlevé, U.F.R. de Mathématique, 59655 Villeneuve d’Ascq, Cedex, France Institut of Mathematics with Computing Centre,Subdivision of the Ufa Research Centre of Russian Academy of Sciences, 45008, Chernyshevsky Str. 112, Ufa, Russia email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that a compact subset of full measure on a generic submanifold of an almost complex manifold is not a pluripolar set. Several related results on boundary behavior of plurisubharmonic functions are established. Our approach is based on gluing a family of complex discs to a generic manifold along a boundary arc and could admit further applications.

Keywords

almost complex structureplurisubharmonic functionpseudoholomorphic disctotally real manifold

MSC classification

Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 2 , October 2020 , pp. 270 - 288
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is partially supported by Labex CEMPI.

References

Audin, M. and Lafontaine, J. (eds.), Holomorphic Curves in Symplectic Geometry, Progress in Mathematics, 117 (Birkhauser, Basel–Boston–Berlin, 1994).10.1007/978-3-0348-8508-9Google Scholar
Bu, S. and Schachermayer, W., ‘Approximation of Jensen measures by image measures under holomorphic functions and applications’, Trans. Amer. Math. Soc. 331 (1992), 585–608.10.1090/S0002-9947-1992-1035999-6Google Scholar
Chirka, E., ‘The Lindelöf and Fatou theorems in ℂn’, Math. USSR Sb. 21 (1973), 619–641.Google Scholar
Coupet, B., Gaussier, H. and Sukhov, A., ‘Some aspects of analysis on almost complex manifolds with boundary’, J. Math. Sci. (N. Y.) 154 (2008), 923–986.10.1007/s10958-008-9202-4Google Scholar
Demailly, J.-P., ‘Complex analytic and differential geometry’, Electronic publication available at http://www-fourier.ujf-grenoble.fr/∼demailly/books.html.Google Scholar
Diederich, K. and Sukhov, A., ‘Plurisubharmonic exhaustion functions and almost complex Stein structures’, Michigan Math. J. 56 (2008), 331–355.10.1307/mmj/1224783517Google Scholar
Harvey, F. R. and Lawson, H. B. Jr, ‘Potential theory on almost complex manifolds’, Ann. Inst. Fourier (Grenoble) 65 (2015), 171–210.10.5802/aif.2928Google Scholar
Khurumov, Y., ‘On the Lindelöf theorem in ℂn’, Dokl. Akad. Nauk SSSR 273 (1983), 1325–1328.Google Scholar
Kuzman, U., ‘Poletsky theory of discs in almost complex manifolds’, Complex Var. Elliptic Equ. 59 (2014), 262–270.10.1080/17476933.2012.734300Google Scholar
Larusson, F. and Sigurdsson, R., ‘Plurisubharmonic functions and analytic discs on manifolds’, J. reine angew. Math. 501 (1998), 1–39.Google Scholar
Levenberg, N., Martin, G. and Poletsky, E., ‘Analytic discs and pluripolar sets’, Indiana Univ. Math. J. 41 (1992), 515–531.Google Scholar
Pinchuk, S., ‘A boundary uniqueness theorem for holomorphic functions of several complex variables’, Mat. Zametki 15 (1974), 205–212.Google Scholar
Poletsky, E., ‘Plurisubharmonic functions as solutions of variational problems’, Proc. Sympos. Pure Math. 52 (1991), 163–171.10.1090/pspum/052.1/1128523Google Scholar
Rosay, J.-P., ‘Approximation of non-holomorphic maps, and Poletsky theory of discs’, J. Korean Math. Soc. 40 (2003), 423–434.10.4134/JKMS.2003.40.3.423Google Scholar
Rosay, J.-P., ‘Pluri-polarity in almost complex structures’, Math. Z. 265 (2010), 133–149.10.1007/s00209-009-0506-yGoogle Scholar
Sadullaev, A., ‘A boundary uniqueness theorem in ℂn’, Mat. Sb. 101 (1976), 568–583.Google Scholar
Sukhov, A., ‘Discs and boundary uniqueness for psh functions on almost complex manifolds’, Ufa Math. J. 10(4) (2018), 129–136.Google Scholar
Sukhov, A. and Tumanov, A., ‘Filling hypersurfaces by discs in almost complex manifolds of dimension 2’, Indiana Univ. Math. J. 57 (2008), 509–544.10.1512/iumj.2008.57.3154Google Scholar
Sukhov, A. and Tumanov, A., ‘Deformation and transversality of pseudoholomorphic discs’, J. Anal. Math. 116 (2012), 1–16.Google Scholar