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}