On a question concerning zero sets of minimal area in domains of [formula omitted

Abstract

Let Ω be a convex domain in C 2 symmetric with respect to the origin and let f( z, w) run over the class of analytic functions on Ω which vanish at the origin. What is the minimal area of a zero set V of such a function f in Ω? In view of known results for the ball, a cube and tube domains, cf. the preceding paper, one might conjecture that the minimum is always attained only if V is a suitable linear variety. The present paper shows, however, that a linear variety V can minimize area only if a rather special analyticity condition is satisfied. The latter condition on Ω and V makes it possible to construct counterexamples to the conjecture.