Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

Abstract

We show that, up to biholomorphism, that there is at most one complete T n -invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold M . We also establish uniqueness without assuming T n -invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra t of T n . As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on CP 1 × C is the standard product metric associated to the Fubini-Study metric on CP 1 and the Euclidean metric on C.

On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

“…This yields (iii). Finally, knowing that the soliton is toric, the uniqueness statement of (iv) is immediate from [Cif20].…”

“…(iii) γ −1 * (JX) is determined and its flow generates a holomorphic isometric S 1 -action of (M, J, γ * g). (iv) Assuming existence, γ * g is the unique T-invariant complete shrinking gradient Kähler-Ricci soliton on M .Conclusions (ii)-(iv) for M = C × P 1 have already been established in [Cif20] where it is shown that any complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on this manifold is isometric to the Cartesian product of the flat Gaussian soliton ω C on C and twice the Fubini-Study metric ω P 1 on P 1 . The new possibility arising is when M is the blowup of C × P 1 at one point, in which case γ −1 * (JX) is given by (2.16).…”

“…Conclusions (ii)-(iv) for M = C × P 1 have already been established in [Cif20] where it is shown that any complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on this manifold is isometric to the Cartesian product of the flat Gaussian soliton ω C on C and twice the Fubini-Study metric ω P 1 on P 1 . The new possibility arising is when M is the blowup of C × P 1 at one point, in which case γ −1 * (JX) is given by (2.16).…”

“…Lemma 2.7) and hence the soliton (up to automorphism) is either that of Feldman-Ilmanen-Knopf [FIK03] on the blowup of C 2 at one point or the flat Gaussian shrinking soliton on C 2 [CDS19]. Here we use a result of [Nab10] to prove, in conjunction with [Cif20], that in the latter case the shrinking soliton is isometric to the cylinder C × P 1 or to a new hypothetical toric shrinking gradient Kähler-Ricci soliton on the blowup of this latter manifold at one point. Being the only possibilities, this allows us to prove a strong form of the Feldman-Ilmanen-Knopf conjecture [FIK03] for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, and in doing so, identify the possible parabolic rescalings that may appear at such singularities.…”

Rigidity of the round cylinders in Ricci shrinkers

“…In higher dimensions, there exist many nontrivial, non-product Ricci shrinkers (e.g., [37] [11] [27]), and the classification of Ricci shrinkers is only achieved when extra assumptions are assumed. Such assumptions include non-negativity of curvatures (e.g., [51] [46][54] [45] [48]), restriction of the Weyl curvatures (e.g., [58][49] [12] [23]), restriction of asymptotic behavior at infinity (e.g., [40] [41]), Kähler conditions (e.g., [55] [22] [20]), and others. In general, much less is known if no extra assumptions are assumed.…”