On Feldman-Ilmanen-Knopf conjecture for the blow-up behavior of the Kahler Ricci flow

Abstract

We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant K\"ahler metric. It is known that the K\"ahler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci flow along the exceptional divisor is the unique $U(n)$-complete shrinking K\"ahler-Ricci soliton on $\mathbb C^n$ blown-up at one point. This establishes the conjecture of Feldman-Ilmanen-Knopf.