The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds

It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.