On small deformations of balanced manifolds

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂∂‾-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂∂‾-degree measuring the difference of Aeppli and Bott-Chern cohomologies with respect to the Betti number b1.