Proceedings of Symposia in Pure Mathematics
Volume 52 (1991), Part 1
Uniqueness for the Dirichlet Problem for
Harmonic Maps from the Annulus
into the Space of Planar Discs
Real hypersurfaces in
defined by equations of the form
(0.1) I m ^ ^ | ^ = 0, a,j&,y,*holomorphic, a3-0y=l,
occur in connection with investigations in one-dimensional function theory
and two-dimensional polynomial convexity. (See [Nev], [AAKl], [G, Chapter
4] as well as the recent string of papers [AW], [BR], [Slodl], [Slod2], [Ber],
If a, P, y, and 3 are defined on a planar domain Q and
(0.2) I m - 0 onQ ,
then equation (0.1) defines a Levi-flat hypersurface 5 in fix C whose fibers
over Q are circles; thus S can also be defined by an equation of the form
(0.3) \w-c(z)\ = r(z).
A hypersurface of the form (0.3) admits a defining equation of the form (0.1)
(locally in Q) if and only if r and c are smooth functions on Q satisfying
the following nonlinear elliptic system of partial differential equations:
= ^ + ^
rczl = 2rzCj, r positive.
Here the subscripts denote differentiation. (See [Ba, Theorem 4] and [Ber,
§3]. The system of equations (0.4) is also equivalent to the identical vanishing
of the Levi-form of the hypersurface (0.3).)
If c = 0 then the system (0.4) just says that u := logr is harmonic,
and suitable functions a, ft, y, and S can be written down with the help
1980 Mathematics Subject Classification (1985 Revision). Primary 32F25, 30E25.
Supported, in part, by a grant from the NSF.
This paper is in final form and no version of it will be submitted for publication elsewhere.
©1991 American Mathematical Society
0082-0717/91 $1.00+ $.25 per page
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