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

DAVID E. BARRETT

Real hypersurfaces in

C2

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],

[Ba].)

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:

(0.4)

rr*

= ^ + ^

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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http://dx.doi.org/10.1090/pspum/052.1/1128509