THE ENVELOPE OF HOLOMORPHY OF A TWO-MANIFOLD IN C2 59
The following theorem is proved by Bishop [I].
Theorem 4.1. The number of elliptic points minus the number of
hyperbolic points equals χ(M), where yfM) denotes the Euler number of M.
Proof. If M is a totally real torus, then M has no exceptional points.
However, the Euler number of M in this case is 0, and the theorem holds.
Therefore assume that M is not totally real. Let G denote the Grassmann
manifold of all oriented two-dimensional real-linear subspaces of C2. By
mapping each point into its oriented tangent plane we obtain the Gauss
map t : M → G.
Using Pliicker coordinates we may identify G with the product of unit
two spheres S1 and S2. Denote by H the subset of G consisting of those
two-dimensional real-linear subspaces of C2 which also have a complex
structure, and whose orientation is induced by this complex structure.
Then p in M is exceptional if and only if t(p)eH or -t(p)εH, where
-t(p) denotes t(p) with orientation reversed. Again using Pliicker coor-
dinates we find that H = (1,0,0) × S2.
We next prove that t (actually an order 2 approximation of t) is trans-
versal to H on M. If t(p)∈7f, by computing a certain determinant we
have sgn(p) = + 1 if p is an elliptic point and sgn(p) = —1 if p is a hyper-
bolic point. If p1,∙∙∙,Pn are the points of M such that t(pl)εH, we define
the intersection number of t(M2) and H as Σ,ifL 1 sgn(pj). Chern and
Spanier [3] have shown that the intersection number is (⅜)χ(Af). By re-
versing the orientation, we find that the intersection number of — t(Λf2)
and H is (⅛)χ(M). Therefore the number of elliptic points minus the number
of hyperbolic points is equal to χ(M). Q.E.D.
Now we are able to prove Theorems 1.1, 1.2, and 1.3.
Proof of Theorem 1.1. From Section 3 we know that the non degenerate
embeddings are an open dense set in the set of embeddings of the two-
sphere in C2. If S2 is a two-sphere embedded in C2 by a non degenerate
embedding, then S2 has at least two elliptic points by Theorem 4.1. Let
Pι,p2,∙∙∙,P( be the elliptic exceptional points. Using Theorem 2.1, we find
that S2 is extendible to S2 U Mi, where Mi is the three-dimensional 4>n,
n ≥ 1, real manifold related to the point pj, i = 1,2,∙∙∙,/. Choosing the
Mi to be disjoint, we set M = = and thus we have that S2 is
extendible to S2UM and Λ∕⊂E(S2). Q.E.D.
The proof of Theorem 1.2 is the same except that we may have only
one exceptional point of the elliptic type.