THE ENVELOPE OF HOLOMORPHY OF A TWO-MANIFOLD IN 55
• I 1 ...
Since =5r∙ =0,pisa critical point in Morse Theory
on ∣p=o oυ lp=o j
(see Milnor [9]). The Hessian matrix is
|
~ o2g(Q) |
fi2g(Q) η |
-2(2)3 + 1) |
0 | |
|
a2g(Q) |
∂2g(0) |
—' |
0 |
2(1 - 2β) |
Thus p is non degenerate if β ≠ ∣ and degenerate if β = ⅜. Hence the
elliptic and hyperbolic points are non degenerate critical points, while the
parabolic points are degenerate critical points. The signature of the matrix
Я is O if Д > ⅜ and 2 if Д < ⅜.
We have similar definitions for elliptic and hyperbolic points in a real
^-dimensional differentiable manifold Mk embedded in Ck.
In [7] the following theorem has been proved.
Theorem 2.1. Let Mk ⊂ Ck be a real к-dimensional differentiable sub-
manifold of Ck such that Mk has at least one exceptional point of the
elliptic type. Then given any n, 1 ≤ n < co, there exists a real (k + 1)-
dimensional 4>n submanifold M such that Mk is extendible to Mk UM.
This theorem was proved by extending the iteration argument devised
by Bishop and applying the analytic continuation result for families of
analytic discs from [22].
Let U be an open neighborhood of the origin in R2 and /be an embedding
of U into R3 ⊂ C3 with /(O) = 0 such that /(0) is a saddle point for
/(L∕). Freeman [2] has shown that f(fJ) is locally polynomially convex.
For example, suppose the equations of f(U) are given by
z1 = и + iv = w
zɪ = u2 — v2.
Then the point и = υ = 0 is a saddle point of f(U), and the intersection
of a closed ball with f(U) is polynomially convex. Thus hyperbolic points
will not in general contribute to the envelope of holomorphy.
Definition 2.4. An exceptional point is called non degenerate if it is
either elliptic or hyperbolic.
Definition 2.5. By a non degenerate embedding we mean an embedding
under which a manifold has at most finitely many exceptional points,
and all such points are non degenerate.