  This paper is devoted to three Gagliardo-Nirenberg-Sobolev interpolation
inequalities on the sphere. We are interested in branches of optimal functions
when a scale parameter varies and investigate whether optimal functions are
constant, or not. In the latter case, a symmetry breaking phenomenon occurs and
our goal is to decide whether the threshold between symmetry and symmetry
breaking is determined by a spectral criterion or not, that is, whether it
appears as a perturbation of the constants or not. The first inequality is
classical while the two other inequalities are variants which reproduce
patterns similar to those observed in Caffarelli-Kohn-Nirenberg inequalities,
for weighted inequalities on the Euclidean space. In the simpler setting of the
sphere, it is possible to implement a parabolic version of the entropy methods
associated to nonlinear diffusion equations, which is so far an open question
on weighted Euclidean spaces.
