Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension > 3

By Y. Siu. 2000

88 citations recorded by the source corpus.

In this paper we prove the following theorem. Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat real hypersurface M in P_n. The condition that M is Levi-flat means that when M is locally defined by the vanishing of a C^m real-valued function f, at every point of M the restriction of d d-bar f to the complex tangent space of M is identically zero. The case of the nonexistence of C^\infty Levi-flat real hypersurface in P_2 is motivated by problems in dynamical systems in P_2.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d380a03701d7ecb90268e2968aaaa59c3d7446e0

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In Applied Mathematics

1. Lihe Wang
2. Jianguo Cao
3. Mei-Chi Shaw
4. Y. Siu
5. Sun-Sig Byun
6. J. Demailly
7. Qi’an Guan
8. Xiangyu Zhou

In Geometry and Topology

1. Lihe Wang
2. Jianguo Cao
3. Mei-Chi Shaw
4. Y. Siu
5. J. Demailly
6. Damian Brotbek
7. Qi’an Guan
8. Mihai Paun

Citations in the corpus

• Estimates for the -Neumann problem and nonexistence of C2 Levi-flat hypersurfaces in (2004, methodology): “Theorem 1 is inspired by two recent papers of Siu [ Siu2 ,Siu3] who proved that there exists no C8 Levi-flat hypersurface in CP n, n ≥ 2. The required smoothness has been reduced to C4 by Iordan [Io].”
• Some recent transcendental techniques in algebraic and complex geometry (2002, background): “The nonexistence of smooth Levi-flat hypersurface in P3 was proved in [19].”