A two-sided H. Lewy extension phenomenon

Authors:
L. R. Hunt and M. Kazlow

Journal:
Proc. Amer. Math. Soc. **74** (1979), 95-100

MSC:
Primary 32D10

DOI:
https://doi.org/10.1090/S0002-9939-1979-0521879-8

MathSciNet review:
521879

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *M* be a real -dimensional CR-manifold in . We are interested in finding conditions on *M* near a point which imply that all CR-function on *M* extend to holomorphic functions in some fixed neighborhood of *p* in . Of course if *M* is a real hypersurface, it is known that *M* having eigenvalues of opposite sign in its Levi form at *p* will give us such an extension result. If we view the Levi form at a point on a general CR-manifold *M* as a quadratic map from the holomorphic tangent space to the normal space of the real tangent space in , and if this map is surjective, then we prove our CR-functions extend to holomorphic functions in an open neighborhood of the point. We also show that if the real codimension of *M* in is 2, and if the Levi form is totally indefinite, then the Levi form is onto as a quadratic map, and hence we have our extension theory.

**[1]**Errett Bishop,*Differentiable manifolds in complex Euclidean space*, Duke Math. J.**32**(1965), 1–21. MR**0200476****[2]**David Ellis, C. Denson Hill, and Chester C. Seabury,*The maximum modulus principle. I. Necessary conditions*, Indiana Univ. Math. J.**25**(1976), no. 7, 709–715. MR**0590086**, https://doi.org/10.1512/iumj.1976.25.25055**[3]**S. J. Greenfield,*Cauchy-Riemann equations in several variables*, Ann. Scuola Norm. Sup. Pisa (3)**22**(1968), 275–314. MR**0237816****[4]**Robert Hermann,*Convexity and pseudoconvexity for complex manifolds*, J. Math. Mech.**13**(1964), 667–672. MR**0167995****[5]**Robert Hermann,*Convexity and pseudoconvexity for complex manifolds*, J. Math. Mech.**13**(1964), 667–672. MR**0167995****[6]**C. Denson Hill and Geraldine Taiani,*Families of analytic discs in 𝐶ⁿ with boundaries on a prescribed CR submanifold*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**5**(1978), no. 2, 327–380. MR**501906****[7]**Lars Hörmander,*An introduction to complex analysis in several variables*, Second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematical Library, Vol. 7. MR**0344507****[8]**L. R. Hunt and R. O. Wells Jr.,*Extensions of CR-functions*, Amer. J. Math.**98**(1976), no. 3, 805–820. MR**0432913**, https://doi.org/10.2307/2373816**[9]**L. R. Hunt and R. O. Wells Jr.,*Holomorphic extension for nongeneric 𝐶𝑅-submanifolds*, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 81–88. MR**0385167****[10]**L. R. Hunt and M. Kazlow,*A two-regular H. Lewy extension phenomenon*(to appear).**[11]**M. Kazlow,*CR functions and tube manifolds*, Trans. Amer. Math. Soc.**255**(1979), 153–171. MR**542875**, https://doi.org/10.1090/S0002-9947-1979-0542875-5**[12]**Isao Naruki,*An analytic study of a pseudo-complex-structure*, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 72–82. MR**0278339****[13]**Ricardo Nirenberg,*On the H. Lewy extension phenomenon*, Trans. Amer. Math. Soc.**168**(1972), 337–356. MR**0301234**, https://doi.org/10.1090/S0002-9947-1972-0301234-4

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
32D10

Retrieve articles in all journals with MSC: 32D10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0521879-8

Keywords:
CR-functions,
Levi form,
quadratic maps,
totally indefinite,
CR-extension

Article copyright:
© Copyright 1979
American Mathematical Society