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Mat. Sb. (N.S.), 1982, Volume 117(159), Number 3, Pages 291–336 (Mi msb2211)  

This article is cited in 34 scientific papers (total in 34 papers)

Regularity of the boundaries of analytic sets

E. M. Chirka


Abstract: In this article the author studies the boundary behavior of a one-dimensional complex analytic set $A$ in a neighborhood of a totally real manifold $M$ in $\mathbf C^n$ with smoothness greater than 1. He proves that the limit points of $A$ on $M$ form a set of locally finite length and that near almost every limit point the closure of $A$ is either a manifold with boundary (with smoothness corresponding to $M$) or a union of two manifolds with boundary. He investigates the structure of the tangent cone to $A$ at the limit points and proves a theorem concerning the boundary regularity of holomorphic discs “glued” to $M$.
Bibliography: 22 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:3, 291–335

Bibliographic databases:

Document Type: Article
UDC: 517.55
MSC: Primary 32C25, 32C40; Secondary 32D99
Received: 01.10.1981

Citation: E. M. Chirka, “Regularity of the boundaries of analytic sets”, Mat. Sb. (N.S.), 117(159):3 (1982), 291–336; Math. USSR-Sb., 45:3 (1983), 291–335

Citation in format AMSBIB
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\by E.~M.~Chirka
\paper Regularity of the boundaries of analytic sets
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 117(159)
\issue 3
\pages 291--336
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=648411}
\zmath{https://zbmath.org/?q=an:0525.32005}
\transl
\jour Math. USSR-Sb.
\yr 1983
\vol 45
\issue 3
\pages 291--335
\crossref{https://doi.org/10.1070/SM1983v045n03ABEH001010}
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  • http://mi.mathnet.ru/eng/msb/v159/i3/p291

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Khurumov I., “The Lindelof Theorem in Space Cn”, 273, no. 6, 1983, 1325–1328  mathscinet  zmath  isi
    2. Jean-Pierre Rosay, “Une remarque a propos de fonctions holomorphes. et du front d'onde C”, Communications in Partial Differential Equations, 12:12 (1987), 1323  crossref
    3. Stefen Hui, “Qualitative properties of solutions to H∞-optimization problems”, Journal of Functional Analysis, 75:2 (1987), 323  crossref
    4. S. I. Pinchuk, S. V. Khasanov, “Asymptotically holomorphic functions and their applications”, Math. USSR-Sb., 62:2 (1989), 541–550  mathnet  crossref  mathscinet  zmath
    5. S. I. Pinchuk, E. M. Chirka, “On the reflection principle for analytic sets”, Math. USSR-Izv., 32:1 (1989), 205–216  mathnet  crossref  mathscinet  zmath
    6. Zbigniew Slodkowski, “Polynomial hulls with convex fibers and complex geodesics”, Journal of Functional Analysis, 94:1 (1990), 156  crossref
    7. Alinhac S., Baouendi M., Rothschild L., “Unique Continuation and Regularity at the Boundary for Holomorphic-Functions”, Duke Math. J., 61:2 (1990), 635–653  crossref  mathscinet  zmath  isi
    8. N. G. Kruzhilin, “Two-dimensional spheres in the boundaries of strictly pseudoconvex domains in $\mathbf C^2$”, Math. USSR-Izv., 39:3 (1992), 1151–1187  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. E. M. Chirka, “Introduction to the geometry of $CR$-manifolds”, Russian Math. Surveys, 46:1 (1991), 95–197  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. A. B. Sukhov, “On algebraicity of complex analytic sets”, Math. USSR-Sb., 74:2 (1993), 419–426  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Bernard Coupet, “Construction de disques analytiques et régularité defonctions holomorphes au bord”, Math Z, 209:1 (1992), 179  crossref  mathscinet  zmath  isi
    12. A. B. Sukhov, “On extension of analytic sets throufh generic manifolds”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 203–212  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. Xiaojun Huang, Steven G. Krantz, “A unique continuation problem for holomorphic mappings”, Communications in Partial Differential Equations, 18:1-2 (1993), 241  crossref
    14. Forstneric F., “A Reflection Principle on Strongly Pseudoconvex Domains with Generic Corners”, Math. Z., 213:1 (1993), 49–64  crossref  mathscinet  zmath  isi
    15. P. A. Soldatkin, “On $H^\infty$-disks attached to completely real manifolds”, Math. Notes, 60:4 (1996), 469–471  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. Josip Globevnik, “Perturbing analytic discs attached to maximal real submanifolds of”, Indagationes Mathematicae, 7:1 (1996), 37  crossref
    17. Yong-Geun Oh, “On the structure of pseudo-holomorphic discs with totally real boundary conditions”, J Geom Anal, 7:2 (1997), 305  crossref
    18. N. G. Karpova, “Removable singularities of plurisubharmonic functions of restricted growth”, Math. Notes, 64:1 (1998), 55–62  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. A. SUKHOV, A. TUMANOV, “STATIONARY DISCS AND GEOMETRY OF CR MANIFOLDS OF CODIMENSION TWO”, Int. J. Math, 12:08 (2001), 877  crossref
    20. Tumanov A., “Extremal Discs and the Regularity of Cr Mappings in Higher Codimension”, Am. J. Math., 123:3 (2001), 445–473  crossref  mathscinet  zmath  isi
    21. Marshall A. Whittlesey, “Polynomial hulls and an optimization problem”, J Geom Anal, 14:3 (2004), 545  crossref
    22. E. M. Chirka, “Variations of Hartogs' Theorem”, Proc. Steklov Inst. Math., 253 (2006), 212–220  mathnet  crossref  mathscinet  elib
    23. LeBrun C., “Twistors, Holornorphic Disks, and Riemann Surfaces with Boundary”, Perspectives in Riemannian Geometry, CRM Proceedings & Lecture Notes, 40, eds. Apostolov V., Dancer A., Hitchin N., Wang M., Amer Mathematical Soc, 2006, 209–221  mathscinet  zmath  isi
    24. Léa Blanc-Centi, “Régularité au bord des applications pseudo-holomorphes propres”, Comptes Rendus Mathematique, 345:8 (2007), 421  crossref
    25. Henkin G., Michel V., “On the Explicit Reconstruction of a Riemann Surface From its Dirichlet-Neumann Operator”, Geom. Funct. Anal., 17:1 (2007), 116–155  crossref  mathscinet  zmath  isi  elib
    26. Bernard Coupet, Hervé Gaussier, Alexandre Sukhov, “Some aspects of analysis on almost complex manifolds with boundary”, Journal of Mathematical Sciences (New York), 154:6 (2008), 923  crossref  elib
    27. Kaushal Verma, “A characterization of domains in C 2 with noncompact automorphism group”, Math Ann, 2008  crossref  zmath  isi
    28. Léa Blanc-Centi, “Regularity and estimates for J-holomorphic discs attached to a maximal totally real submanifold”, Journal of Mathematical Analysis and Applications, 341:1 (2008), 170  crossref
    29. LeBrun C., Mason L.J., “The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks”, Math. Res. Lett., 16:2-3 (2009), 291–301  mathscinet  zmath  isi
    30. Arne Schneck, “Constrained Hardy space approximation”, Journal of Approximation Theory, 162:8 (2010), 1466  crossref
    31. Shafikov R., Verma K., “Holomorphic Mappings Between Domains in C-2”, Can. J. Math.-J. Can. Math., 64:2 (2012), 429–454  crossref  mathscinet  zmath  isi
    32. Florian Bertrand, Léa Blanc-Centi, “Stationary holomorphic discs and finite jet determination problems”, Math. Ann, 2013  crossref
    33. Shafikov R., Sukhov A., “Local Polynomial Convexity of the Unfolded Whitney Umbrella in C-2”, Int. Math. Res. Notices, 2013, no. 22, 5148–5195  crossref  isi
    34. S. Pinchuk, R. Shafikov, A. Sukhov, “Some Aspects of Holomorphic Mappings: A Survey”, Proc. Steklov Inst. Math., 298 (2017), 212–247  mathnet  crossref  crossref  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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