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Complex Analysis On boundaries of Levi-flat hypersurfaces in Cn

Summary: Complex Analysis
On boundaries of Levi-flat hypersurfaces in Cn
Pierre Dolbeault a
, Giuseppe Tomassini b
, Dmitri Zaitsev c
aInstitut de math´ematiques de Jussieu, universit´e Paris 6, 175, rue du Chevaleret, 75013 Paris, France
bScuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
cSchool of Mathematics, Trinity College, Dublin 2, Ireland
Received 16 June 2005; accepted after revision 19 July 2005
Presented by Paul Malliavin
Let S be a smooth 2-codimensional real compact submanifold of Cn
, n > 2. We address the problem of finding a
compact hypersurface M, with boundary S, such that M \S is Levi-flat. We prove the following theorem. Assume
that (i) S is nonminimal at every CR point, (ii) every complex point of S is flat and elliptic and there exists
at least one such point, (iii) S does not contain complex submanifolds of dimension n - 2. Then there exists a
Levi-flat (2n - 1)-subvariety fM C × Cn
with negligible singularities and boundary eS (in the sense of currents)
such that the natural projection : C × Cn


Source: Abbondandolo, Alberto - Scuola Normale Superiore of Pisa


Collections: Mathematics