52
RICE UNIVERSITY STUDIES
morph y of the compact set K, and is denoted by E(K) (see [6] where this
definition is seen to be equivalent to an earlier one given in [72]).
The main result of this paper is the following theorem which will be
proved in Section 4.
Theorem 1.1. There is an open dense set U of embeddings of the
real two-sphere in C2 such that each embedded two-sphere from U has
the property that its envelope of holomorphy contains a real three-di-
mensional cSn manifold Af (1 ≤ n < co).
(Outline of Proof). Using an iteration technique devised by Bishop [7],
we establish a (/<—l)-ρarameter family of analytic discs in a neighborhood
of an elliptic point (to be defined in Section 2) in a real ∕<-dimensional
differentiable submanifold Mk of Ck. By computing a certain Jacobian
and using a theorem on simultaneous analytic continuation from [77],
we are able to say that every function holomorphic on Mk can be extended
to a holomorphic function on a real (fc + l)-dimensional fffπ manifold
(1 ≤ n < oo).
Using Thom Transversality Theory (see Levine [5]), we prove that there
exists an open dense set of embeddings of any compact two-manifold in
C2 so that each such embedded manifold has the following properties:
(i) There are at most finitely many exceptional points, and
(ii) Each exceptional point is of the elliptic or hyperbolic type.
(The concepts of exceptional points and hyperbolic points, as well as elliptic
points, will be defined in Section 2. They are first and second order con-
ditions on the submanifold at a point).
Bishop [7] proves that for such a two-manifold the number of elliptic
points minus the number of hyperbolic points equals the Euler number
of the two-manifold. In particular, each two-sphere, embedded by an
element in the open dense set, has at least two elliptic points. Using the
analytic continuation result for Mk ɑ Ck we can easily complete the proof.
Remark: Considerthe two-sphere in standard position in R2 ɑ C2
(R3 — {(z1,z2)εC2: Imz2 = θ})∙ Using a classical argument involving the
Cauchy integral formula (cf. Bochner-Martin [2]), we can say that every
function holomorphic on the two-sphere can be extended to a function
holomorphic in the open ball. In this case the envelope of holomorphy
of the two-sphere is the closed unit ball. This is an example of an embedded
two-sphere which has the property that its envelope of holomorphy con-
tains a three-manifold, but does not contain a manifold of higher dimension.
A manifold M embedded in C' is said to be totally real if there does