The name is absent



172                 RICE UNIVERSITY STUDIES

borhood of such a point is holomorphically convex. On the other hand,
evidence is presented here which indicates that sufficiently small compact
neighborhoods of a “hyperbolic” point on a two-manifold in
C2 are holo-
morphically convex.

The terms “elliptic” and “hyperbolic” describe two cases which arise
after the following normalization pointed out by Bishop [7, p. 4]. Near
any point on a two-manifold in C2 it may be represented as the graph of
a smooth complex-valued function of two real variables. After a trans-
lation which takes this point to the origin in
C2, the manifold is coincident
near zero with the graph of

(1.1)                /(ɪ) = az + bz + <2(z) + o(∣z∣2),

where Q is a quadratic form in z and z with complex coefficients. Now the
holomorphic convexity of sets in the manifold near 0 is invariant under
biholomorphic coordinate changes leaving 0 fixed. Denoting the usual
coordinates of
C2 by (z, w), the particular map (z, w)→(z, w — az) reduces
the representation (1.1) to one in which
a = 0, so that this restriction may
henceforth be assumed. If
b ≠ 0, then it can be shown [7] that there is
a disc Δ centered at 0 in C and a domain of holomorphy U containing
{(z,∕(z)):
z ∈Δ} such that the latter set is convex with respect to the
functions holomorphic on
U. The problem thus remains only for the case
b = 0, in which

(1.2)                         ∕ = <2 + κ

where R(z) = o(∣z∣2). By an elementary biholomorphic coordinate change
[I] which does not affect the conditions /(0) =
a = b = 0, Bishop showed
how to convert (1.2) into a representation of the same form in which
Q is a
reα∕-valucd quadratic form. This condition may thus be added to (1.2).

Assume now that zero is not an eigenvalue of Q. The given point in M is
elliptic if both eigenvalues of Q have the same sign and hyperbolic if they
have different signs. The conclusions of Theorems 1.1 and 1.3 and certain
other examples suggest that small discs in
M centered at a hyperbolic
point are holomorphically convex.

If / happens to be real, this result is a consequence of a deep theorem
due to Mergelyan.

Theorem 1.1 [7 J. If f is a real valued continuous function on a closed
disc
Δ such that for each number β the level set

(1.3)


Lβ = {z∈∆√(z) = β}




More intriguing information

1. Party Groups and Policy Positions in the European Parliament
2. The name is absent
3. Behaviour-based Knowledge Systems: An Epigenetic Path from Behaviour to Knowledge
4. A Rare Case Of Fallopian Tube Cancer
5. Peer Reviewed, Open Access, Free
6. Howard Gardner : the myth of Multiple Intelligences
7. Sex differences in the structure and stability of children’s playground social networks and their overlap with friendship relations
8. The name is absent
9. Electricity output in Spain: Economic analysis of the activity after liberalization
10. Non-farm businesses local economic integration level: the case of six Portuguese small and medium-sized Markettowns• - a sector approach