Mathematical Research Letters

Volume 23 (2016)

Number 6

On Feldman–Ilmanen–Knopf’s conjecture for the blow-up behavior of the Kähler Ricci flow

Pages: 1681 – 1719

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n6.a6

Authors

Bin Guo (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.; and Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Jian Song (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Abstract

We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is proved in [9, 22, 32] that the Kähler–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 Kähler–Ricci soliton on $\mathbb{C}^n$ blown-up at one point. This establishes the conjecture of Feldman–Ilmanen–Knopf [8].

Full Text (PDF format)