Using a nonlinear parabolic flow, in this talk I will explain why the optimal regions of symmetry and symmetry breaking for the extremals of critical and subcritical Caffarelli-Kohn-Nirenberg inequalities are related to the spectral gap of the linearized problem around the asymptotic Barenblatt solutions. This is a surprising result since it means that a global test yields a global result. The use of the parabolic flow also allows to get improved inequalities with explicit remainder terms.