CR regular embeddings of 𝑆⁴ⁿ⁻¹ in ℂ²ⁿ⁺¹

Kasuya, Naohiko

doi:10.1090/proc/14962

CR regular embeddings of $S^{4n-1}$ in $\mathbb {C}^{2n+1}$
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by Naohiko Kasuya

Proc. Amer. Math. Soc. 148 (2020), 3021-3024

DOI: https://doi.org/10.1090/proc/14962

Published electronically: March 17, 2020

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Abstract:

Ahern and Rudin have given an explicit construction of a totally real embedding of $S^3$ in $\mathbb {C}^3$. As a generalization of their example, we give an explicit example of a CR regular embedding of $S^{4n-1}$ in $\mathbb {C}^{2n+1}$. Consequently, we show that the odd dimensional sphere $S^{2m-1}$ with $m>1$ admits a CR regular embedding in $\mathbb {C}^{m+1}$ if and only if $m$ is even.

References

Patrick Ahern and Walter Rudin, Totally real embeddings of $S^3$ in $\textbf {C}^3$, Proc. Amer. Math. Soc. 94 (1985), no. 3, 460–462. MR 787894, DOI 10.1090/S0002-9939-1985-0787894-7
Michèle Audin, Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagragiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), no. 4, 593–623 (French). MR 966952, DOI 10.1007/BF02566781
Elie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl. 11 (1933), no. 1, 17–90 (French). MR 1553196, DOI 10.1007/BF02417822
Ali M. Elgindi, On the non-existence of CR-regular embeddings of $S^5$, Complex Var. Elliptic Equ. 64 (2019), no. 9, 1564–1567. MR 3976437, DOI 10.1080/17476933.2018.1542690
Franc Forstnerič, On totally real embeddings into $\textbf {C}^n$, Exposition. Math. 4 (1986), no. 3, 243–255. MR 880125
M. L. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 221–225. MR 0420697
M. L. Gromov, Convex integration of differential relations. I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343 (Russian). MR 0413206
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
Howard Jacobowitz and Peter Landweber, Manifolds admitting generic immersions into $\Bbb C^N$, Asian J. Math. 11 (2007), no. 1, 151–165. MR 2304588, DOI 10.4310/AJM.2007.v11.n1.a14
Naohiko Kasuya and Masamichi Takase, Generic immersions and totally real embeddings, Internat. J. Math. 29 (2018), no. 11, 1850073, 12. MR 3871721, DOI 10.1142/S0129167X18500738
Naohiko Kasuya and Masamichi Takase, Erratum: Generic immersions and totally real embeddings, Internat. J. Math. 30 (2019), no. 12, 1992003, 2. MR 4039107, DOI 10.1142/s0129167x19920034
Hon Fei Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. Amer. Math. Soc. 172 (1972), 1–33. MR 314066, DOI 10.1090/S0002-9947-1972-0314066-8
J. Alexander Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976), no. 2, 217–224. MR 410764
Marko Slapar, Cancelling complex points in codimension two, Bull. Aust. Math. Soc. 88 (2013), no. 1, 64–69. MR 3096869, DOI 10.1017/S0004972712000652
Marko Slapar, CR regular embeddings and immersions of compact orientable 4-manifolds into $\Bbb {C}^3$, Internat. J. Math. 26 (2015), no. 5, 1550033, 5. MR 3345510, DOI 10.1142/S0129167X15500330
Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan 14 (1962), 397–429. MR 145555, DOI 10.2969/jmsj/01440397
Noboru Tanaka, On generalized graded Lie algebras and geometric structures. I, J. Math. Soc. Japan 19 (1967), 215–254. MR 221418, DOI 10.2969/jmsj/01920215
Rafael Torres, CR regular embeddings and immersions of 6-manifolds into complex 4-space, Proc. Amer. Math. Soc. 144 (2016), no. 8, 3493–3498. MR 3503716, DOI 10.1090/proc/13034
Rafael Torres, An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space, Enseign. Math. 63 (2017), no. 1-2, 165–180. MR 3777134, DOI 10.4171/LEM/63-1/2-5
R. O. Wells Jr., Holomorphic hulls and holomorphic convexity, Rice Univ. Stud. 54 (1968), no. 4, 75–84. MR 241690
R. O. Wells Jr., Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann. 179 (1969), 123–129. MR 237823, DOI 10.1007/BF01350124