The Oka principle for tame families of Stein manifolds

Let $X$ be a smooth open manifold of even dimension, $T$ be a topological space, and $\mathcal{J}=\{J_t{\}}_{t\in T}$ be a continuous family of smooth integrable Stein structures on $X$. Under suitable additional assumptions on $T$ and $\mathcal{J}$, we prove an Oka principle for continuous families of maps from the family of Stein manifolds $(X,J_t)$, $t\in T$, to any Oka manifold, showing that every family of continuous maps is homotopic to a family of $J_t$-holomorphic maps depending continuously on $t$. We also prove the Oka-Weil theorem for sections of $\mathcal{J}$-holomorphic vector bundles on $Z=T\times X$ and the Oka principle for isomorphism classes of such bundles. The assumption on the family $\mathcal{J}$ is that the $J_t$-convex hulls of any compact set in $X$ are upper semicontinuous with respect to $t\in T$; such a family is said to be tame. For suitable parameter spaces $T$, we characterise tameness by the existence of a continuous family ${\rho }_t:X\to {\mathbb{R}}_{+}=[0,+\mathrm{\infty })$, $t\in T$, of strongly $J_t$-plurisubharmonic exhaustion functions on $X$. Every family of complex structures on an open orientable surface is tame.We give an example of a nontame smooth family of Stein structures $J_t$ on ${\mathbb{R}}^{2n}$ $(t\in \mathbb{R},n>1)$ such that $({\mathbb{R}}^{2n},J_t)$ is biholomorphic to ${\mathbb{C}}^n$ for every $t\in \mathbb{R}$. We show that the Oka principle fails on any nontame family.