Perelman's entropy functional at type I singularities of the Ricci flow

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber (2010) and Enders-Müller-Topping (2011) that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit W-density at a Type I singular point to the entropy of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.

We study blow--ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Mueller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit W-density at a Type I singular point to the "entropy" of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.