COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES
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초록
Let M be a C∞ real hypersurface in Cn+1, n ¸ 1, locally given as the zero locus of a C∞ real valued function r that is defined on a neighborhood of the reference point P ∈ M. For each k = 1, . . . , n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n − k at P. The problem is to find an integral manifold of the real 1-form i□r on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.
Abstract
Let M be a C∞ real hypersurface in Cn+1, n ¸ 1, locally given as the zero locus of a C∞ real valued function r that is defined on a neighborhood of the reference point P ∈ M. For each k = 1, . . . , n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n − k at P. The problem is to find an integral manifold of the real 1-form i□r on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.
참고문헌 (16)
- A. Boggess, "CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics" (1991), CRC Press
- R. Bryant, "Exterior Differential Systems" (1986), Springer-Verlag
- S. S. Chern, "Real hypersurfaces in complex manifolds", Acta Math., 133, pp.219~271 (1974)
- G. Darboux, "Sur le probl`eme de Pfaff (1), (2)", Bull.Sci. Math., 6, pp.14~36 (1882)
- K. Diederich, "Pseudoconvex domains with real-analytic boundary", Ann. Math., 107(2), pp.371~384 (1978)
- N. G. Dieu, "Real hypersurface containing complex varieties, preprint", ~
- G. Frobenius, "¨ Uber das Pfaffsche probleme", J. Reine und Angew. Math., 82, pp.230~315 (1877)
- P. Griffiths, "Differential Systems and Isometric Embeddings, Annals of Mathematics Studies, 114. The William H. Roever Lectures in Geometry" (1987), Princeton University Press
- 한종규, "Solvability of overdetermined PDE systems that admit a complete prolongation and some local problems in CR geometry", 대한수학회지, 40(4), pp.695~708 (2003), 대한수학회
- C. K. Han, "Pfaffian systems of Frobenius type and solvability of generic overdetermined PDE systems, Symmetries and overdetermined systems of partial differential equations", Math. Appl., 144, pp.421~429 (2008), Springer
- 한종규, "Generalization of the Frobenius theorem on involutivity", 대한수학회지, 46(5), pp.1087~1103 (2009), 대한수학회
- M. Kuranishi, "On E. Cartan’s prolongation theorem of exterior differential systems", Amer. J. Math., 79, pp.1~47 (1957)
- H. Lewy, "On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables", Ann. of Math., 64, pp.514~522 (1956)
- J. M. Tr´epreau, "The holomorphic extension of CR functions defined on a real C2 hypersurface in Cn", Invent. Math., 83(3), pp.583~592 (1986)
- A. Tumanov, "Extension of CR-functions into a wedge", Mat. Sb., 181(7), pp.951~964 (1990)
- S. Wang, "Exterior differential system" (2005), Lectures at Seoul Nat. Univ. based on R. Bryant’s lectures at Duke Univ