  A foliated manifold (M,F) is minimizable if there exists a Riemannian metric
g on M such that every leaf of F is a minimal submanifold of (M,g). Alvarez
Lopez defined a cohomology class of degree 1 called the Alvarez class of (M,F)
whose triviality characterizes the minimizability of (M,F), when M is closed
and F is Riemannian. In this paper, we show that the family of the Alvarez
classes of a smooth family of Riemannian foliations is continuous with respect
to the parameter. Since the Alvarez class has algebraic rigidity under certain
topological conditions on (M,F) as the author showed in arXiv:0909.1125, we
show that the minimizability of Riemannian foliations is invariant under
deformation under the same topological conditions.
