Holomorphic curves in Lorentzian CR-manifolds,
Trans. Amer. Math. Soc., vol. 272 no. 1
(1982),
pp. 203–221 [MR83i:32029]
(last updated on 2004/04/06)
Abstract:
When can a real hypersurface in complex
n-space contain any complex
curves? Since the tangent spaces to such a curve
would have to be null
vectors for the Levi form, a necessary condition is that
the Levi form
have zeros. The simplest way this can happen in the
non-degenerate case
is for the Levi form to have the Lorentzian signature.
In this paper, I show that a Lorentzian CR-manifold
M has at most a
finite parameter family of holomorphic curves, in fact, at
most an n2
parameter family if the dimension of M is 2n+1. This
maximum is attained,
as I show by example. When n=2, the only way it can be
reached is for M
to be CR-flat. In higher dimensions, where the CR-flat
model does not achieve
the maximum, it is still unknown whether or not there is
more than one
local model with the maximal dimension family of
holomorphic curves.
The technique used is exterior differential systems
together with the
Chern-Moser theory in the n=2 case.
Reprints are available, but can also be downloaded
from the AMS or from JSTOR
| 
Mathematics Department