CHAPTER 1
CR Structures
§1. Some observations of Poincare. This book studies the induced structure
on real hypersurfaces M
n+l
in the complex space
C"+1
and a generaliza-
tion to abstract manifolds. Most of the phenomena and difficulties already
occur for n = 1 ; we generally restrict ourselves to this case. The subject
arose from a paper of Poincare in which he considered whether the Riemann
Mapping Theorem could be generalized from C to C . He gave two simple
arguments that even a local version, which is trivial in C , does not hold in
higher dimensions and he also laid the basis for a global counterexample.
This chapter starts with his local arguments; the next with his global one.
Any two real analytic curves in
C1
are locally equivalent: Given points p
and q on the curves T{ and T2 there are open subsets of
C1
, U{ containing
p and U2 containing q, and a biholomorphism I: U{ U2 with 0( Ux n
Y{) = U2 D T2. This may be taken as a very weak form of the Riemann
Mapping Theorem. Poincare showed that the analogous result does not hold
in C . Namely, let s and S be real analytic surfaces of real dimension
three in C
2
. In general, there will not be a local biholomorphism taking
one to the other. The first of Poincare's two proofs of this uses the fact that
a function on a hypersurface is the restriction of a holomorphic function
only if it satisfies a certain partial differential equation. We give Poincare's
derivation of this fact (using mostly his notation). Let the hypersurface be
written as a graph
s = {(Xj + iy{, x2 + iy2): x{ = /(y{, x2, y2)}
and let F(y{, x2, y2) be the function on s. We ask: Does there exist a
holomorphic function /(z
1
, z2) of zx = xx + iy{ and z2 = x2 + iy2 such
that F{yx, x2, y2) = f(t{yx, x2, v2) + iyx, x2 + iy2) ? If so
dyx dz{
i
dz2 dz{
-2
If this notation is not familiar, look at the next section where it is formally
introduced. Let
(1)
d
L = 0_ -
-: dvx
,
,
d
•vi
OZ1
1
http://dx.doi.org/10.1090/surv/032/01
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