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.