On Lewy extension for smooth hypersurfaces in $\mathbb {C}^n \times \mathbb {R}$
Authors:
Jiří Lebl, Alan Noell and Sivaguru Ravisankar
Journal:
Trans. Amer. Math. Soc. 371 (2019), 6581-6603
MSC (2010):
Primary 32V40; Secondary 32V25
DOI:
https://doi.org/10.1090/tran/7605
Published electronically:
October 24, 2018
MathSciNet review:
3937338
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset \mathbb {C}^{n}\times \mathbb {R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of $\mathbb {C}^n \times \mathbb {R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.
- Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
- Valentin Burcea, A normal form for a real 2-codimensional submanifold in $\Bbb {C}^{N+1}$ near a CR singularity, Adv. Math. 243 (2013), 262–295. MR 3062747, DOI https://doi.org/10.1016/j.aim.2013.04.018
- Valentin Burcea, On a family of analytic discs attached to a real submanifold $M\subset \Bbb C^{N+1}$, Methods Appl. Anal. 20 (2013), no. 1, 69–78. MR 3085782, DOI https://doi.org/10.4310/MAA.2013.v20.n1.a4
- David Catlin, Boundary behavior of holomorphic functions on pseudoconvex domains, J. Differential Geometry 15 (1980), no. 4, 605–625 (1981). MR 628348
- Adam Coffman, CR singularities of real fourfolds in $\Bbb C^3$, Illinois J. Math. 53 (2009), no. 3, 939–981 (2010). MR 2727363
- Pierre Dolbeault, Giuseppe Tomassini, and Dmitri Zaitsev, On boundaries of Levi-flat hypersurfaces in $\Bbb C^n$, C. R. Math. Acad. Sci. Paris 341 (2005), no. 6, 343–348 (English, with English and French summaries). MR 2169149, DOI https://doi.org/10.1016/j.crma.2005.07.012
- Pierre Dolbeault, Giuseppe Tomassini, and Dmitri Zaitsev, Boundary problem for Levi flat graphs, Indiana Univ. Math. J. 60 (2011), no. 1, 161–170. MR 2952414, DOI https://doi.org/10.1512/iumj.2011.60.4241
- Hanlong Fang and Xiaojun Huang, Flattening a non-degenerate CR singular point of real codimension two, Geom. Funct. Anal. 28 (2018), no. 2, 289–333. MR 3788205, DOI https://doi.org/10.1007/s00039-018-0431-5
- Xiang Hong Gong, Normal forms of real surfaces under unimodular transformations near elliptic complex tangents, Duke Math. J. 74 (1994), no. 1, 145–157. MR 1271467, DOI https://doi.org/10.1215/S0012-7094-94-07407-3
- Xianghong Gong and Jiří Lebl, Normal forms for CR singular codimension-two Levi-flat submanifolds, Pacific J. Math. 275 (2015), no. 1, 115–165. MR 3336931, DOI https://doi.org/10.2140/pjm.2015.275.115
- Monique Hakim and Nessim Sibony, Spectre de $A(\bar \Omega )$ pour des domaines bornés faiblement pseudoconvexes réguliers, J. Functional Analysis 37 (1980), no. 2, 127–135 (French, with English summary). MR 578928, DOI https://doi.org/10.1016/0022-1236%2880%2990037-3
- Gary Alvin Harris, The traces of holomorphic functions on real submanifolds, Trans. Amer. Math. Soc. 242 (1978), 205–223. MR 477120, DOI https://doi.org/10.1090/S0002-9947-1978-0477120-1
- C. D. Hill and G. Taiani, On the Hans Lewy extension phenomenon in higher codimension, Proc. Amer. Math. Soc. 91 (1984), no. 4, 568–572. MR 746091, DOI https://doi.org/10.1090/S0002-9939-1984-0746091-0
- Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
- Xiao Jun Huang and Steven G. Krantz, On a problem of Moser, Duke Math. J. 78 (1995), no. 1, 213–228. MR 1328757, DOI https://doi.org/10.1215/S0012-7094-95-07809-0
- Xiaojun Huang and Wanke Yin, A Bishop surface with a vanishing Bishop invariant, Invent. Math. 176 (2009), no. 3, 461–520. MR 2501295, DOI https://doi.org/10.1007/s00222-008-0167-1
- Xiaojun Huang and Wanke Yin, A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric, Int. Math. Res. Not. IMRN 15 (2009), 2789–2828. MR 2525841, DOI https://doi.org/10.1093/imrn/rnp033
- Xiaojun Huang and Wanke Yin, Flattening of CR singular points and analyticity of the local hull of holomorphy I, Math. Ann. 365 (2016), no. 1-2, 381–399. MR 3498915, DOI https://doi.org/10.1007/s00208-015-1228-6
- Xiaojun Huang and Wanke Yin, Flattening of CR singular points and analyticity of the local hull of holomorphy II, Adv. Math. 308 (2017), 1009–1073. MR 3600082, DOI https://doi.org/10.1016/j.aim.2016.12.008
- 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, DOI https://doi.org/10.1007/BF01393370
- Jiří Lebl, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang, CR singular images of generic submanifolds under holomorphic maps, Ark. Mat. 52 (2014), no. 2, 301–327. MR 3255142, DOI https://doi.org/10.1007/s11512-013-0193-0
- Jiří Lebl, Alan Noell, and Sivaguru Ravisankar, Extension of CR functions from boundaries in $\Bbb C^n\times \Bbb R$, Indiana Univ. Math. J. 66 (2017), no. 3, 901–925. MR 3663330, DOI https://doi.org/10.1512/iumj.2017.66.6067
- Jiří Lebl, Alan Noell, and Sivaguru Ravisankar, Codimension two CR singular submanifolds and extensions of CR functions, J. Geom. Anal. 27 (2017), no. 3, 2453–2471. MR 3667437, DOI https://doi.org/10.1007/s12220-017-9767-6
- B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
- Jürgen Moser, Analytic surfaces in ${\bf C}^2$ and their local hull of holomorphy, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 397–410. MR 802502, DOI https://doi.org/10.5186/aasfm.1985.1044
- Jürgen K. Moser and Sidney M. Webster, Normal forms for real surfaces in ${\bf C}^{2}$ near complex tangents and hyperbolic surface transformations, Acta Math. 150 (1983), no. 3-4, 255–296. MR 709143, DOI https://doi.org/10.1007/BF02392973
- B. V. Shabat, Introduction to complex analysis. Part II, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. Joel. MR 1192135
- Marko Slapar, On complex points of codimension 2 submanifolds, J. Geom. Anal. 26 (2016), no. 1, 206–219. MR 3441510, DOI https://doi.org/10.1007/s12220-014-9545-7
- A. E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type, Mat. Sb. (N.S.) 136(178) (1988), no. 1, 128–139 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 129–140. MR 945904, DOI https://doi.org/10.1070/SM1989v064n01ABEH003298
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32V40, 32V25
Retrieve articles in all journals with MSC (2010): 32V40, 32V25
Additional Information
Jiří Lebl
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
MR Author ID:
813143
ORCID:
0000-0002-9320-0823
Email:
lebl@math.okstate.edu
Alan Noell
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
MR Author ID:
131880
Email:
noell@math.okstate.edu
Sivaguru Ravisankar
Affiliation:
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bengaluru 560065, India
MR Author ID:
1054138
Email:
sivaguru@tifrbng.res.in
Keywords:
Extension of CR functions,
Lewy extension,
CR singularity,
Levi-flat
Received by editor(s):
August 18, 2017
Received by editor(s) in revised form:
February 1, 2018
Published electronically:
October 24, 2018
Additional Notes:
The first author was supported in part by NSF grant DMS-1362337.
Article copyright:
© Copyright 2018
American Mathematical Society