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)



Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of $ {\bf C}\sp{n}$

Author: Al Boggess
Journal: Trans. Amer. Math. Soc. 272 (1982), 351-374
MSC: Primary 32F20; Secondary 32F25, 35N15
MathSciNet review: 656494
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The relationship between the Levi geometry of a submanifold of $ {{\mathbf{C}}^n}$ and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of $ {{\mathbf{C}}^n}$, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in $ {{\mathbf{C}}^n}$. This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of $ {{\mathbf{C}}^n}$. In fact, we show that if $ S$ is a real analytic, generic, submanifold of $ {{\mathbf{C}}^n}$ (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then $ S$ is not extendible to any open set in $ {{\mathbf{C}}^n}$.

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

  • [AB] A. Boggess, Kernels for the tangential Cauchy Riemann equations, Ph.D. thesis, Rice Univ., May, 1979; Trans. Amer. Math. Soc. 262 (1980), 1-50. MR 583846 (82b:32030)
  • [C] E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1945.
  • [F] H. Flanders, Differential forms with applications to the physical sciences, Academic Press, New York, 1963. MR 0162198 (28:5397)
  • [G] R. B. Gardner, Invariants of Pfaffian systems, Trans. Amer. Math. Soc. 126 (1967), 514-533. MR 0211352 (35:2233)
  • [Gr] S. J. Greenfield, Cauchy Riemann equations in several variables, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 22 (1968), 275-314. MR 0237816 (38:6097)
  • [H] G. M. Henkin, The H. Lewy equation and analysis of pseudo-convex manifolds, Uspehi Mat. Nauk 32 (1977), 57-118; Russian Math. Surveys 32 (1977), 59-130. MR 0454067 (56:12318)
  • [Ho] L. Hörmander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 0203075 (34:2933)
  • [HP] R. Harvey and J. Polking, Fundamental solutions in complex analysis. I, II, Duke Math. J. 46 (1979), 253-300, 301-340.
  • [HT] C. D. Hill and G. Taiani, Families of analytic discs in $ {{\mathbf{C}}^n}$ with boundaries on a prescribed $ CR$-submanifold (preprint).
  • [HW] L. R. Hunt and R. O. Wells, Extension of $ CR$-functions, Amer. J. Math. 98 (1976), 805-820. MR 0432913 (55:5892)
  • [N] R. Nirenberg, On the H. Lewy extension phenomenon, Trans. Amer. Math. Soc. 168 (1972), 337-356. MR 0301234 (46:392)
  • [NW] R. Nirenberg and R. O. Wells, Approximation theorems on differentiable submanifolds of a complex manifold, Trans. Amer. Math. Soc. 142 (1969), 15-25. MR 36 #7140. MR 0245834 (39:7140)
  • [T] G. Tomassini, Trace delle funzioni olomorphe sulle sorrovarieta analytiche reali d'una varieta complessa, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 20 (1966), 31-43. MR 0206992 (34:6808)
  • [W] R. O. Wells, Jr., Holomorphic hulls and holomorphic convexity of differentiable submanifolds, Trans. Amer. Math. Soc. 132 (1968), 245-262. MR 0222340 (36:5392)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F20, 32F25, 35N15

Retrieve articles in all journals with MSC: 32F20, 32F25, 35N15

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society