  This paper is motivated by the characterization of the optimal symmetry
breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence,
optimal functions and sharp constants are computed in the symmetry region. The
result solves a longstanding conjecture on the optimal symmetry range.
  As a byproduct of our method we obtain sharp estimates for the principal
eigenvalue of Schr\"odinger operators on some non-flat non-compact manifolds,
which to the best of our knowledge are new.
  The method relies on generalized entropy functionals for nonlinear diffusion
equations. It opens a new area of research for approaches related to carr\'e du
champ methods on non-compact manifolds. However key estimates depend as much on
curvature properties as on purely nonlinear effects. The method is well adapted
to functional inequalities involving simple weights and also applies to general
cylinders. Beyond results on symmetry and symmetry breaking, and on optimal
constants in functional inequalities, rigidity theorems for nonlinear elliptic
equations can be deduced in rather general settings.
