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)

 
 

 

On the regularity of CR mappings between CR manifolds of hypersurface type


Authors: S. Berhanu and Ming Xiao
Journal: Trans. Amer. Math. Soc. 369 (2017), 6073-6086
MSC (2010): Primary 32V05, 32V10, 32V20; Secondary 32H02, 32H40
DOI: https://doi.org/10.1090/tran/6818
Published electronically: May 11, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove smooth and analytic versions of the classical Schwarz reflection principle for transversal CR mappings between two Levi-nondegenerate CR manifolds of hypersurface type.


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

  • [BER] M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103
  • [BJT] M. S. Baouendi, H. Jacobowitz, and F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985), no. 2, 365-400. MR 808223, https://doi.org/10.2307/1971307
  • [BHu] M. S. Baouendi and Xiaojun Huang, Super-rigidity for holomorphic mappings between hyperquadrics with positive signature, J. Differential Geom. 69 (2005), no. 2, 379-398. MR 2169869
  • [Be] Eric Bedford, Proper holomorphic mappings, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 157-175. MR 733691, https://doi.org/10.1090/S0273-0979-1984-15235-2
  • [BN] Steven R. Bell and Raghavan Narasimhan, Proper holomorphic mappings of complex spaces, Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer, Berlin, 1990, pp. 1-38. MR 1095089
  • [BCH] Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. MR 2397326
  • [BX] S. Berhanu and Ming Xiao, On the $ C^\infty $ version of the reflection principle for mappings between CR manifolds, Amer. J. Math. 137 (2015), no. 5, 1365-1400. MR 3405870, https://doi.org/10.1353/ajm.2015.0034
  • [CKS] Joseph A. Cima, Steven G. Krantz, and Ted J. Suffridge, A reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications, Math. Z. 186 (1984), no. 1, 1-8. MR 735046, https://doi.org/10.1007/BF01215486
  • [CS] J. A. Cima and T. J. Suffridge, A reflection principle with applications to proper holomorphic mappings, Math. Ann. 265 (1983), no. 4, 489-500. MR 721883, https://doi.org/10.1007/BF01455949
  • [CGS] Bernard Coupet, Hervé Gaussier, and Alexandre Sukhov, Regularity of CR maps between convex hypersurfaces of finite type, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3191-3200. MR 1610940, https://doi.org/10.1090/S0002-9939-99-04908-4
  • [DW] K. Diederich and S. M. Webster, A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), no. 4, 835-843. MR 596117
  • [EH] Peter Ebenfelt and Xiaojun Huang, On a generalized reflection principle in $ \mathbb{C}^2$, Complex analysis and geometry (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 125-140. MR 1912734
  • [EL] Peter Ebenfelt and Bernhard Lamel, Finite jet determination of CR embeddings, J. Geom. Anal. 14 (2004), no. 2, 241-265. MR 2051686, https://doi.org/10.1007/BF02922071
  • [E] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • [Fe] Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. MR 0350069
  • [Fr1] Franc Forstnerič, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), no. 1, 31-61. MR 969413, https://doi.org/10.1007/BF01394144
  • [Fr2] Franc Forstnerič, Proper holomorphic mappings: a survey, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 297-363. MR 1207867
  • [Hu1] Xiao Jun Huang, On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 433-463 (English, with English and French summaries). MR 1296739
  • [Hu2] Xiaojun Huang, Geometric analysis in several complex variables, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)-Washington University in St. Louis. MR 2691738
  • [KP] S. I. Pinchuk and S. V. Khasanov, Asymptotically holomorphic functions and their applications, Mat. Sb. (N.S.) 134(176) (1987), no. 4, 546-555, 576 (Russian); English transl., Math. USSR-Sb. 62 (1989), no. 2, 541-550. MR 933702
  • [La1] Bernhard Lamel, A $ C^\infty $-regularity theorem for nondegenerate CR mappings, Monatsh. Math. 142 (2004), no. 4, 315-326. MR 2085046, https://doi.org/10.1007/s00605-003-0040-7
  • [La2] Bernhard Lamel, A reflection principle for real-analytic submanifolds of complex spaces, J. Geom. Anal. 11 (2001), no. 4, 627-633. MR 1861300, https://doi.org/10.1007/BF02930759
  • [La3] Bernhard Lamel, Holomorphic maps of real submanifolds in complex spaces of different dimensions, Pacific J. Math. 201 (2001), no. 2, 357-387. MR 1875899, https://doi.org/10.2140/pjm.2001.201.357
  • [Le] H. Lewy, On the boundary behavior of holomorphic mappings, Acad. Naz. Lincei, 3, 1-8 (1977).
  • [M] Nordine Mir, An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings, Math. Z. 231 (1999), no. 1, 189-202. MR 1696763, https://doi.org/10.1007/PL00004723
  • [NWY] L. Nirenberg, S. Webster, and P. Yang, Local boundary regularity of holomorphic mappings, Comm. Pure Appl. Math. 33 (1980), no. 3, 305-338. MR 562738, https://doi.org/10.1002/cpa.3160330306
  • [Pi] S. I. Pinčuk, Holomorphic mappings of real-analytic hypersurfaces, Mat. Sb. (N.S.) 105(147) (1978), no. 4, 574-593, 640 (Russian). MR 496595
  • [R] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923
  • [T] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459
  • [Tu] A. Tumanov, Analytic discs and the regularity of CR mappings in higher codimension, Duke Math. J. 76 (1994), no. 3, 793-807. MR 1309333, https://doi.org/10.1215/S0012-7094-94-07633-3
  • [W] S. M. Webster, On mapping an $ n$-ball into an $ (n+1)$-ball in complex spaces, Pacific J. Math. 81 (1979), no. 1, 267-272. MR 543749

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32V05, 32V10, 32V20, 32H02, 32H40

Retrieve articles in all journals with MSC (2010): 32V05, 32V10, 32V20, 32H02, 32H40


Additional Information

S. Berhanu
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: berhanu@temple.edu

Ming Xiao
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

DOI: https://doi.org/10.1090/tran/6818
Received by editor(s): December 14, 2014
Received by editor(s) in revised form: August 19, 2015
Published electronically: May 11, 2017
Additional Notes: The work of the first author was supported in part by NSF DMS 1300026
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society