Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

TL;DR

This work examines the finite-time evolution of unstable conformal perturbations of the Fubini–Study metric on $\\mathbb{CP}^2$ under Ricci flow in four dimensions. Using a cohomogeneity-one, $U(2)$-symmetric setup and the Ricci–DeTurck trick, the authors simulate the flow from small non-Kähler perturbations and observe locally collapsing behavior that, after parabolic rescaling, converges to the blowdown soliton $\\mathcal{L}^2_{-1}$, with the evolving region becoming asymptotically Kähler but with reversed complex structure. The numerical evidence—finite-time singularity formation, convergence of the cone angle to the $\\mathcal{L}^2_{-1}$ cone, and the approach to $K=-1$ near the distinguished fiber—supports the conjectured hierarchy in CHI04 that unstable FS perturbations yield singularities modeled by $\\mathcal{L}^2_{-1}$. These results illuminate non-Kähler-to-Kähler singularity formation and the role of parabolic dilations in identifying soliton models for Ricci-flow blowups.

Abstract

Kröncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Krö20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].

Figures (3)

• Figure 1: $1/{\kappa_{23}}$ plotted vs. time
• Figure 2: $K=g{g_s}/f$ plotted vs. $\theta$ near the final time
• Figure 3: $\gamma^2 = f^2/g^2$ plotted vs. length near the final time

Theorems & Definitions (1)

• Remark