On the contact between complex manifolds and real hypersurfaces in $\textbf {C}^{3}$
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- by Thomas Bloom PDF
- Trans. Amer. Math. Soc. 263 (1981), 515-529 Request permission
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 $\sup \{ m \in {\mathbf {Z}}|$ 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
- Thomas Bloom and Ian Graham, A geometric characterization of points of type $m$ on real submanifolds of $\textbf {C}^{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182. MR 492369
- Thomas Bloom and Ian Graham, On “type” conditions for generic real submanifolds of $\textbf {C}^{n}$, Invent. Math. 40 (1977), no. 3, 217–243. MR 589930, DOI 10.1007/BF01425740
- Thomas Bloom, Remarks on type conditions for real hypersurfaces in $\textbf {C}^{n}$, Several complex variables (Cortona, 1976/1977) Scuola Norm. Sup. Pisa, Pisa, 1978, pp. 14–24. MR 681297
- Thomas Bloom, Sur le contact entre sous-variétés réelles et sous-variétés complexes de $\textbf {C}^{n}$, Séminaire Pierre Lelong (Analyse) (année 1975/76), Lecture Notes in Math., Vol. 578, Springer, Berlin, 1977, pp. 28–43 (French). MR 0589904 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). J. D’Angelo, Finite type conditions for real hypersurfaces, J. Differential Geometry (to appear).
- Makhlouf Derridj, Estimations pour $\bar \partial$ dans des domaines non pseudo-convexes, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 239–254, xi (French, with English summary). MR 513888, DOI 10.5802/aif.723
- Klas Diederich and John E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 107 (1978), no. 2, 371–384. MR 477153, DOI 10.2307/1971120
- Michael Freeman, Integration of analytic differential systems with singularities and some applications to real submanifolds of $\textbf {C}^{n}$, J. Math. Soc. Japan 30 (1978), no. 3, 571–578. MR 590087, DOI 10.2969/jmsj/03030571
- Roe W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Mathematics, Vol. 562, Springer-Verlag, Berlin-New York, 1976. MR 0442149, DOI 10.1007/BFb0087594
- Peter Greiner, Subelliptic estimates for the $\bar \delta$-Neumann problem in $C^{2}$, J. Differential Geometry 9 (1974), 239–250. MR 344702
- J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
- J. J. Kohn, Subellipticity of the $\bar \partial$-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), no. 1-2, 79–122. MR 512213, DOI 10.1007/BF02395058 —, 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.
- J. J. Kohn, Sufficient conditions for subellipticity on weakly pseudo-convex domains, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 6, 2214–2216. MR 466635, DOI 10.1073/pnas.74.6.2214
- Steven G. Krantz, Characterizations of various domains of holomorphy via $\bar \partial$ estimates and applications to a problem of Kohn, Illinois J. Math. 23 (1979), no. 2, 267–285. MR 528563
- Hugo E. Rossi, Differentiable manifolds in complex euclidean space, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 512–516. MR 0234499
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
Additional Information
- © Copyright 1981 American Mathematical Society
- 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