E. E. Levi convexity and the Hans Lewy problem. Part I : reduction to vanishing theorems
Suppose that is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space to is a uniform limit of entire maps . We prove that a holomorphic map from a closed complex subvariety in a Stein manifold admits a holomorphic extension provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.