Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Polynomially convex hulls of singular real manifolds


Authors: Rasul Shafikov and Alexandre Sukhov
Journal: Trans. Amer. Math. Soc. 368 (2016), 2469-2496
MSC (2010): Primary 32E20, 32E30, 32V40, 53D12
DOI: https://doi.org/10.1090/tran/6422
Published electronically: July 10, 2015
MathSciNet review: 3449245
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of $ \mathbb{C}^n$ with self-intersections and open Whitney umbrella points.


References [Enhancements On Off] (What's this?)

  • [1] H. Alexander, On the totally real spheres of Ahern and Rudin and Weinstein, The Madison Symposium on Complex Analysis (Madison, WI, 1991) Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 29-35. MR 1190967 (93m:32020), https://doi.org/10.1090/conm/137/1190967
  • [2] H. Alexander, Gromov's method and Bennequin's problem, Invent. Math. 125 (1996), no. 1, 135-148. MR 1389963 (97j:32007), https://doi.org/10.1007/s002220050071
  • [3] V. I. Arnold, Symplectic geometry and topology, J. Math. Phys. 41 (2000), no. 6, 3307-3343. MR 1768639 (2001h:53108), https://doi.org/10.1063/1.533315
  • [4] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. Volume 1, Classification of critical points, caustics and wave fronts; translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012. Reprint of the 1985 edition. MR 2896292
  • [5] Eric Bedford and Wilhelm Klingenberg, On the envelope of holomorphy of a $ 2$-sphere in $ {\bf C}^2$, J. Amer. Math. Soc. 4 (1991), no. 3, 623-646. MR 1094437 (92j:32034), https://doi.org/10.2307/2939272
  • [6] Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5-42. MR 972342 (89k:32011)
  • [7] Th. Bröcker, Differentiable germs and catastrophes, translated from the German, last chapter and bibliography by L. Lander; London Mathematical Society Lecture Note Series, No. 17, Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR 0494220 (58 #13132)
  • [8] E. M. Chirka, Regularity of the boundaries of analytic sets, Mat. Sb. (N.S.) 117(159) (1982), no. 3, 291-336, 431 (Russian). MR 648411 (83f:32009)
  • [9] Evgeni M. Chirka, Bernard Coupet, and Alexandre B. Sukhov, On boundary regularity of analytic discs, Michigan Math. J. 46 (1999), no. 2, 271-279. MR 1704142 (2000f:32022), https://doi.org/10.1307/mmj/1030132410
  • [10] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999 (38 #325)
  • [11] J.-P. Demailly, Complex analytic and differential geometry, electronic publication available at http://www-fourier.ujf-grenoble.fr/$ \sim $demailly/books.html.
  • [12] Tien-Cuong Dinh and Mark G. Lawrence, Polynomial hulls and positive currents, Ann. Fac. Sci. Toulouse Math. (6) 12 (2003), no. 3, 317-334 (English, with English and French summaries). MR 2030090 (2005b:32022)
  • [13] Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53-106. MR 0650816 (58 #31276)
  • [14] Freddy Dumortier, Jaume Llibre, and Joan C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006. MR 2256001 (2007f:34001)
  • [15] Freddy Dumortier, Paulo R. Rodrigues, and Robert Roussarie, Germs of diffeomorphisms in the plane, Lecture Notes in Mathematics, vol. 902, Springer-Verlag, Berlin-New York, 1981. MR 653474 (83f:58008)
  • [16] Julien Duval and Nessim Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487-513. MR 1344768 (96f:32016), https://doi.org/10.1215/S0012-7094-95-07912-5
  • [17] Julien Duval and Nessim Sibony, Hulls and positive closed currents, Duke Math. J. 95 (1998), no. 3, 621-633. MR 1658760 (2000a:32015), https://doi.org/10.1215/S0012-7094-98-09515-1
  • [18] John Erik Fornæss and Nessim Sibony, Riemann surface laminations with singularities, J. Geom. Anal. 18 (2008), no. 2, 400-442. MR 2393266 (2009b:32042), https://doi.org/10.1007/s12220-008-9018-y
  • [19] Franc Forstnerič and Jean-Pierre Rosay, Approximation of biholomorphic mappings by automorphisms of $ {\bf C}^n$, Invent. Math. 112 (1993), no. 2, 323-349. MR 1213106 (94f:32032), https://doi.org/10.1007/BF01232438
  • [20] Franc Forstnerič, A theorem in complex symplectic geometry, J. Geom. Anal. 5 (1995), no. 3, 379-393. MR 1360826 (96j:32038), https://doi.org/10.1007/BF02921802
  • [21] Franc Forstneric, Actions of $ (\mathbf {R},+)$ and $ (\mathbf {C},+)$ on complex manifolds, Math. Z. 223 (1996), no. 1, 123-153. MR 1408866 (97i:32041), https://doi.org/10.1007/PL00004552
  • [22] Franc Forstnerič, Stein manifolds and holomorphic mappings, The homotopy principle in complex analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. MR 2975791
  • [23] F. Forstnerič and E. L. Stout, A new class of polynomially convex sets, Ark. Mat. 29 (1991), no. 1, 51-62. MR 1115074 (92h:32016), https://doi.org/10.1007/BF02384330
  • [24] A. B. Givental, Lagrangian imbeddings of surfaces and the open Whitney umbrella, Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 35-41, 96 (Russian). MR 868559 (88g:58018)
  • [25] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518 (49 #6269)
  • [26] Sushil Gorai, Local polynomial convexity of the union of two totally-real surfaces at their intersection, Manuscripta Math. 135 (2011), no. 1-2, 43-62. MR 2783386 (2012g:32018), https://doi.org/10.1007/s00229-010-0405-x
  • [27] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307-347. MR 809718 (87j:53053), https://doi.org/10.1007/BF01388806
  • [28] M. Gromov, Soft and hard symplectic geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 81-98. MR 934217 (89f:53059)
  • [29] Go-o Ishikawa, Symplectic and Lagrange stabilities of open Whitney umbrellas, Invent. Math. 126 (1996), no. 2, 215-234. MR 1411129 (97h:58022), https://doi.org/10.1007/s002220050095
  • [30] S. Ivashkovich and V. Shevchishin, Reflection principle and $ J$-complex curves with boundary on totally real immersions, Commun. Contemp. Math. 4 (2002), no. 1, 65-106. MR 1890078 (2002k:32047), https://doi.org/10.1142/S0219199702000592
  • [31] B. Jöricke, Local polynomial hulls of discs near isolated parabolic points, Indiana Univ. Math. J. 46 (1997), no. 3, 789-826. MR 1488338 (98j:32005), https://doi.org/10.1512/iumj.1997.46.1501
  • [32] Carlos E. Kenig and Sidney M. Webster, The local hull of holomorphy of a surface in the space of two complex variables, Invent. Math. 67 (1982), no. 1, 1-21. MR 664323 (84c:32014), https://doi.org/10.1007/BF01393370
  • [33] François Lalonde and Jean-Claude Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991), no. 1, 18-33 (French). MR 1090163 (92f:58060), https://doi.org/10.1007/BF02566634
  • [34] Evgeny A. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993), no. 1, 85-144. MR 1218708 (94c:32007), https://doi.org/10.1512/iumj.1993.42.42006
  • [35] Rasul Shafikov and Alexandre Sukhov, Local polynomial convexity of the unfolded Whitney umbrella in $ \mathbb{C}^2$, Int. Math. Res. Not. IMRN 22 (2013), 5148-5195. MR 3129096
  • [36] S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866. MR 0185604 (32 #3067)
  • [37] Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474 (2008d:32012)
  • [38] Dmitry Turaev, Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps, Nonlinearity 16 (2003), no. 1, 123-135. MR 1950779 (2003j:37085), https://doi.org/10.1088/0951-7715/16/1/308
  • [39] Barnet M. Weinstock, On the polynomial convexity of the union of two maximal totally real subspaces of $ {\bf C}^n$, Math. Ann. 282 (1988), no. 1, 131-138. MR 960837 (89i:32035), https://doi.org/10.1007/BF01457016
  • [40] Erlend Fornæss Wold, A note on polynomial convexity: Poletsky disks, Jensen measures and positive currents, J. Geom. Anal. 21 (2011), no. 2, 252-255. MR 2772071 (2011m:32013), https://doi.org/10.1007/s12220-010-9130-7

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32E20, 32E30, 32V40, 53D12

Retrieve articles in all journals with MSC (2010): 32E20, 32E30, 32V40, 53D12


Additional Information

Rasul Shafikov
Affiliation: Department of Mathematics, The University of Western Ontario, London, Ontario, N6A 5B7, Canada
Email: shafikov@uwo.ca

Alexandre Sukhov
Affiliation: Université des Sciences et Technologies de Lille, U.F.R. de Mathématiques, 59655 Villeneuve d’Ascq, Cedex, France
Email: sukhov@math.univ-lille1.fr

DOI: https://doi.org/10.1090/tran/6422
Keywords: Symplectic structure, totally real manifold, Lagrangian manifold, Whitney umbrella, polynomial convexity, analytic disc, characteristic foliation
Received by editor(s): September 3, 2013
Received by editor(s) in revised form: January 20, 2014
Published electronically: July 10, 2015
Additional Notes: The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada
The second author was partially supported by Labex CEMPI
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society