  We construct inhomogeneous isoparametric families of hypersurfaces with
non-austere focal set on each symmetric space of non-compact type and rank
greater than or equal to 3. If the rank is greater than or equal to 4, there
are infinitely many such examples. Our construction yields the first examples
of isoparametric families on any Riemannian manifold known to have a
non-austere focal set. They can be obtained from a new general extension method
of submanifolds from Euclidean spaces to symmetric spaces of non-compact type.
This method preserves the mean curvature and isoparametricity, among other
geometric properties.
