Polynomial hulls with convex fibers and complex geodesics

Abstract

Let X be a compact subset of {| z | = 1} × C n with convex fibers. Several equivalent conditions are obtained that characterize polynomially convex hulls Y = X ̂ with nonempty interior. Under the latter assumption, it is shown that every ( z 0 , w 0 ) ϵ ∂ Y and such that ¦z 0 ¦ < 1 is contained in a closed n -dimensional complex submanifold of d × C n which is tangent to Y along an analytic disc. We show, as an application, that some results of Lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls.