Interpolation inequalities on the sphere: linear vs. nonlinear flows

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.