Curvature estimates for minimal submanifolds of higher codimension and small G-rank

We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\text{ir}n+m,m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean spaces of arbitrary codimension.