The name is absent

LOCAL HOLOMORPHIC CONVEXITY OF A TWO-MANIFOLD INO 179

Proof. It will suffice to prove that h is zero on a small open ball B
centered at O. If H is any plane in R³ through the (-axis, it is obvious that
there is a sequence {l_n} of positive real numbers such that t„ → O and the
point

P_π = (<_B,g(t_n)sin(l/Q, g(t_n) cos (l∕t_n))

is in H Г» B for each n. Up to an isomorphism of H with R², {p,,} and
h∖H ∩B satisfy Lemma 3.1. Therefore h ∣ H C∖ B = 0. Since H is ar-
bitrary, h I B = O.

Now let

k(t) = s(O∞s(l∕t) -I- ⅛(t)sin(l∕t), t ≠ O,

and F be any real-analytic function in two real variables.

Theorem 3.3. If φ is holomorphic in a connected neighborhood of
O in C² and takes real values on

M = {(z,F(z) + fc(∣z[)): z ≠ 0}

then φ is constant.

If f is any infinitely differentiable function on a neighborhood of O in C
and f_m is the partial sum of degree m of the Taylor series for f, it follows
that ∕₀(z) = ∕_m(z) + fc( I z I ) admits no non constant holomorphic functions
taking real values on its graph, and /(z) — ∕₀(z) = o(∣z∣^m). In this sense,
examples of this behavior are “dense.” Note: It is not asserted that an
f₀ with this property can be found arbitrarily close to f in the C°^c topology
of some neighborhood of O. This may be true, but no proof is offered here.

Proof. As noted already, it is enough to show that M is a determining
set for real-analytic functions. Since the map

(z, w) → (z, w — F(z))

is a real-analytic coordinate change which carries M onto

M' = {(z,fc(∣ zI)): z ≠ 0},

it will suffice that M' is a determining set for real-analytic functions of four
real variables.

Suppose h is such a function defined on an open ball B centered at O.
If h I M' = O, z ≠ O is any point of C, and K_z is the hyperplane

K_z = {tz: t real} × C,

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