  This paper is devoted to rigidity results for some elliptic PDEs and related
interpolation inequalities of Sobolev type on smooth compact connected
Riemannian manifolds without boundaries. Rigidity means that the PDE has no
other solution than the constant one at least when a parameter is in a certain
range. This parameter can be used as an estimate for the best constant in the
corresponding interpolation inequality. Our approach relies in a nonlinear flow
of porous medium / fast diffusion type which gives a clear-cut interpretation
of technical choices of exponents done in earlier works. We also establish two
integral criteria for rigidity that improve upon known, pointwise conditions,
and hold for general manifolds without positivity conditions on the curvature.
Using the flow, we are also able to discuss the optimality of the corresponding
constant in the interpolation inequalities.
