UNIQUENESS FOR THE DIRICHLET PROBLEM 11
Pick a local defining function for S as in (0.1), and let / be the local
holomorphic function whose graph is the leaf of St passing through (zQ9w).
Let
a{z)f{z) + fi(z)
n[Z)
y(z)f(z) + 6{z)'
Then ImA(z) 0 and ImA(z0) = 0 so that h is constant, and an ana-
lytic continuation argument shows that the entire leaf of St passing through
(z0, w) is a subset of S. But following this leaf out to the boundary of Q
we contradict the earlier observation that St n (bSl x C) lies strictly inside
Sn(bQxC). n
Appendix 2. The extended Dirichlet problem is not always solvable. In this
Appendix we give an example, adapted from one shown to the author by
Forstneric, of a smooth hypersurface with circular fibers in bQ x C, with
Q = the unit disk, which is not the boundary of a Levi-flat hypersurface in
Q x C with (generalized) circular fibers.
Let v be the meromorphic SL(2, C)-valued function v(z) = I +
z~lN,
where N is the constant matrix \{) J
x
) . We claim that there is no SL(2, (re-
valued function \i, continuous on CI and holomorphic in Q, such that
S(/i) n (M2 x C) = 5(i/) n (ba x C).
Indeed, if such a function JU exists, then the SL(2, C)-valued function
£ := ii -
v~X
is continuous on Q with boundary values in SL(2, R), and
holomorphic in Q with at worst a simple pole at the origin. Let M2 be the
(matrix-valued) residue of £ atO. Then
£(z)-z~lM2-zM2
is holomorphic
on Q with boundary values in SL(2, R) and hence constant. Thus £(z) =
Mx +
z~lM2
+ zM2 with Mx e SL(2, R).
To guarantee that /u £, v has no pole at the origin, we must have
M2N = 0 = M{N + M2. It follows that /i(z) = Mx(I - ~NN - z~N), so that
1 = det//(0) = det M
r
det(/-iVW), contradicting the fact that det(/-AW) =
0. Thus no function JLL satisfies the above requirements.
The reader may check that the above example S(v) is gauge-equivalent to
the hypersurface \w - l/z| = 1. Our conclusions here can also be deduced
(perhaps more insightfully) from the notion of the "index" of a leaf used in
[BG] in [Forst2].
NOTE ADDED IN PROOF.
The maps studied in this paper are not har-
monic in the usual differential-geometric sense. The most natural differential-
geometric interpretation of the notion of a harmonic map from a Riemann
surface into the target space consisting of discs in C makes use of the
SL(2, C)-invariant Lorentz metric
(dr2
-
\dc\2)/r2.
But regardless of the
choice of target metric the corresponding system of partial differential equa-
tions will be invariant under conjugation in z and thus cannot match (0.4).
The complex-analytic justification for the terminology of the title still stands.
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