The name is absent

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(p_l)εH, we define
the intersection number of t(M²) and H as Σ,_i^fL ₁ sgn(p_j). 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 M_i, where M_i is the three-dimensional 4>ⁿ,
n ≥ 1, real manifold related to the point p_j, i = 1,2,∙∙∙,/. Choosing the
M_i 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.

<<   6   7   8   9   10   11   12   >>

More intriguing information