  This paper is devoted to sharp interpolation inequalities on the sphere and
their proof using flows. The method explains some rigidity results and proves
uniqueness in related semilinear elliptic equations. Nonlinear flows allow to
cover the interval of exponents ranging from Poincar\'e to Sobolev inequality,
while an intriguing limitation (an upper bound on the exponent) appears in the
carr\'e du champ method based on the heat flow. We investigate this limitation,
describe a counter-example for exponents which are above the bound, and obtain
improvements below.
