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