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)

 
 

 

On the contact between complex manifolds and real hypersurfaces in $ {\bf C}\sp{3}$


Author: Thomas Bloom
Journal: Trans. Amer. Math. Soc. 263 (1981), 515-529
MSC: Primary 32F30; Secondary 53B35
DOI: https://doi.org/10.1090/S0002-9947-1981-0594423-0
MathSciNet review: 594423
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ m$ be a real $ {\mathcal{C}^\infty }$ hypersurface of an open subset of $ {{\mathbf{C}}^3}$ and let $ p \in M$. Let $ {a^1}(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $ p$ in $ {{\mathbf{C}}^3}$ with $ M$ at $ p$. Let $ {c^1}(M,p)$ denote the $ \sup \{ m \in {\mathbf{Z}}\vert$ for all tangential holomorphic vector fields $ L$ with $ L(p) \ne 0$ then $ {L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak{L}_M}(L))(p) = 0\} $ where $ {i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}$ are positive integers such that $ \sum\nolimits_{t = 0}^n {{i_t} + {j_t} = m - 3} $ and $ {\mathfrak{L}_M}(L)$ denotes the Levi form of $ M$ evaluated on the vector field $ L$.

Theorem. If $ M$ is pseudoconvex near $ p \in M$ then $ {a^1}(M,p) = {c^1}(M,p)$.


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

  • [1] T. Bloom and I. Graham, A geometric characterization of points of type $ m$ on real submanifolds of $ {{\mathbf{C}}^n}$, J. Differential Geometry 12 (1977), 171-182. MR 0492369 (58:11495)
  • [2] -, On 'type' conditions for generic real submanifolds of $ {{\mathbf{C}}^n}$, Invent. Math. 40 (1977), 217-243. MR 0589930 (58:28644)
  • [3] T. Bloom, Remarks on type conditions for real hypersurfaces in $ {{\mathbf{C}}^n}$ (Proc. Internat. Conf. on Several Complex Variables, Cortona, Italy, 1976-77), Scuola Norm. Sup. Pisa, Pisa, 1978, pp. 14-24. MR 681297 (84f:32022)
  • [4] -, Sur le contact entre sous-variétés réelles et sous-variétés complexes de $ {{\mathbf{C}}^n}$, Séminaire Pierre Lelong, 1975/76, Lecture Notes in Math., no. 578, Springer-Verlag, Berlin and New York, 1977, pp. 28-43. MR 0589904 (58:28643)
  • [5] D. Catlin, Necessary conditions for subellipticity and hypoellipticity for the $ \bar \partial $-Neumann problem on pseudoconvex domains (Proc. Conf. Several Complex Variables, Princeton, N. J., 1979) (to appear).
  • [6] J. D'Angelo, Finite type conditions for real hypersurfaces, J. Differential Geometry (to appear).
  • [7] M. Derridj, Estimations pour $ \bar \partial $ dans des domaines non pseudo-convexes, Ann. Inst. Fourier (Grenoble) 28 (1978), 239-254. MR 513888 (80b:32021)
  • [8] K. Diederich and J.-E. Fornaess, Pseudoconvex domains with real analytic boundary, Ann. of Math. (2) 107 (1978), 371-384. MR 0477153 (57:16696)
  • [9] M. Freeman, Integration of analytic differential systems with singularities and some applications to real submanifolds of $ {{\mathbf{C}}^n}$, J. Math. Soc. Japan 30 (1978), 571-578. MR 0590087 (58:28675)
  • [10] R. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Math., no. 562, Springer-Verlag, Berlin and New York, 1976. MR 0442149 (56:537)
  • [11] P. C. Greiner, On subelliptic estimates for the $ \bar \partial $-Neumann problem in $ {{\mathbf{C}}^2}$, J. Differential Geometry 9 (1974), 239-250. MR 0344702 (49:9441)
  • [12] J. J. Kohn, Boundary behaviour of $ \bar \partial $ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523-542. MR 0322365 (48:727)
  • [13] -, Subellipticity of the $ \bar \partial $-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), 79-122. MR 512213 (80d:32020)
  • [14] -, Subellipticity of the $ \bar \partial $-Neumann problem on weakly pseudo-convex domains, Rencontres sur l'Analyse Complexe à Plusieurs Variables et les Systèmes Indéterminés, Université de Montréal Press, Montréal, 1975, pp. 105-118.
  • [15] -, Sufficient conditions for subellipticity on weakly pseudo-convex domains, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 2214-2216. MR 0466635 (57:6512)
  • [16] S. G. Krantz, Characterizations of various domains of holomorphy via $ \bar \partial $ estimates and applications to a problem of Kohn, Illinois J. Math. 23 (1979), 267-285. MR 528563 (80h:32036)
  • [17] H. Rossi, Differentiable manifolds in complex Euclidean space (Proc. Internat. Congr. Math., Moscow, 1966), "Mir", Moscow, 1968, pp. 512-516. MR 0234499 (38:2816)
  • [18] L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. MR 0436223 (55:9171)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F30, 53B35

Retrieve articles in all journals with MSC: 32F30, 53B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0594423-0
Keywords: $ \bar \partial $-Neumann problem, pseudoconvex hypersurface, tangential holomorphic vector field, type of a point
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society