The Oka principle for tame families of Stein manifolds

Let X X be a smooth open manifold of even dimension, T T be a topological space, and J = { J t } t ∈ T J=\{J_t\}t∈ T be a continuous family of smooth integrable Stein structures on X X . Under suitable additional assumptions on T T and J J , we prove an Oka principle for continuous families of maps from the family of Stein manifolds ( X , J t ) (X,J_t) , t ∈ T t∈ T , to any Oka manifold, showing that every family of continuous maps is homotopic to a family of J t J_t -holomorphic maps depending continuously on t t . We also prove the Oka–Weil theorem for sections of J J -holomorphic vector bundles on Z = T × X Z=T× X and the Oka principle for isomorphism classes of such bundles. The assumption on the family J J is that the J t