  We introduce a fractional Yamabe flow involving nonlocal conformally
invariant operators on the conformal infinity of asymptotically hyperbolic
manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it
converges to the standard sphere up to a M\"obius diffeomorphism. This result
allows us to obtain extinction profiles of solutions of some fractional porous
medium equations. In the end, we use this fractional fast diffusion equation,
together with its extinction profile and some estimates of its extinction time,
to improve a Sobolev inequality via a quantitative estimate of the remainder
term.
