A two-sided H. Lewy extension phenomenon
HTML articles powered by AMS MathViewer
- by L. R. Hunt and M. Kazlow PDF
- Proc. Amer. Math. Soc. 74 (1979), 95-100 Request permission
Abstract:
Let M be a ${C^\infty }$ real $(n + k)$-dimensional CR-manifold in ${{\mathbf {C}}^n}$. We are interested in finding conditions on M near a point $p \in M$ which imply that all CR-function on M extend to holomorphic functions in some fixed neighborhood of p in ${{\mathbf {C}}^n}$. 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 ${{\mathbf {C}}^n}$, 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 ${{\mathbf {C}}^n}$ is 2, and if the Levi form is totally indefinite, then the Levi form is onto ${{\mathbf {R}}^2}$ as a quadratic map, and hence we have our extension theory.References
- Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
- 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 590086, DOI 10.1512/iumj.1976.25.25055
- S. J. Greenfield, Cauchy-Riemann equations in several variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 275–314. MR 237816
- Robert Hermann, Convexity and pseudoconvexity for complex manifolds, J. Math. Mech. 13 (1964), 667–672. MR 0167995
- Robert Hermann, Convexity and pseudoconvexity for complex manifolds, J. Math. Mech. 13 (1964), 667–672. MR 0167995
- C. Denson Hill and Geraldine Taiani, Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 327–380. MR 501906
- Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Mathematical Library, Vol. 7, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0344507
- L. R. Hunt and R. O. Wells Jr., Extensions of CR-functions, Amer. J. Math. 98 (1976), no. 3, 805–820. MR 432913, DOI 10.2307/2373816
- L. R. Hunt and R. O. Wells Jr., Holomorphic extension for nongeneric $\textrm {CR}$-submanifolds, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 81–88. MR 0385167 L. R. Hunt and M. Kazlow, A two-regular H. Lewy extension phenomenon (to appear).
- M. Kazlow, CR functions and tube manifolds, Trans. Amer. Math. Soc. 255 (1979), 153–171. MR 542875, DOI 10.1090/S0002-9947-1979-0542875-5
- Isao Naruki, An analytic study of a pseudo-complex-structure, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. Tokyo Press, Tokyo, 1970, pp. 72–82. MR 0278339
- Ricardo Nirenberg, On the H. Lewy extension phenomenon, Trans. Amer. Math. Soc. 168 (1972), 337–356. MR 301234, DOI 10.1090/S0002-9947-1972-0301234-4
Additional Information
- © Copyright 1979 American Mathematical Society
- 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