A fractional Yamabe flow and some applications

Understand

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.