Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples

Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb{T}}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $${\lambda }_s<1<{\lambda }_c<{\lambda }_u\mathrm{.}$$ Under the assumption that the set $$\{x:\mid \log \det (Tf{\mid }_{E^{cu}(x)})\mid \le \log {\lambda }_u\}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu}=E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particular, $f$ is robustly transitive.\par We build, with this criterion, a new example of a $C^1$ open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.