Linear stability of the blowdown Ricci shrinker in 4D

TL;DR

The authors prove that the 4-dimensional blowdown shrinking Ricci soliton of Feldman–Ilmanen–Knopf is strictly linearly stable in the Cao–Hamilton–Ilmanen sense. They develop a detailed tensor harmonic analysis adapted to the weighted Lichnerowicz Laplacian $L_f = \Delta_f + 2\mathcal{R}$, leveraging the soliton’s $U(2)$-invariance and Kähler/self-dual structure. They show that nonnegative eigen-directions of $L_f$ are precisely the Ricci tensor and gauge transformations, and that all other perturbations decay under the linearized flow, after appropriate gauge fixing. The proof splits into a radial (I) part, tackled by reducing to a small finite-dimensional subspace and applying Barta-type estimates, and a nonradial (II) part, handled via a Wigner-function decomposition on Berger-sphere cross-sections, with high-frequency modes controlled by negative zeroth-order parts and remaining low-frequency modes analyzed through finite-dimensional blocks and Sturm–Liouville arguments. This establishes the blowdown soliton as the first known non-cylindrical linearly stable shrinking soliton in 4D, supporting the conjectural picture that Ricci flow generically performs blowdowns and related topological surgeries in four dimensions.

Abstract

We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups. The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its $U(2)$-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a $9$-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations.

Figures (17)

• Figure 1: Plot of $F(r)$ with $R = 1$.
• Figure 2: The maximum eigenvalue of $r^2 \mathcal{Q}^J(r)$ (as a function of $r$) for $J \in \{0,1,2,3,4,5\}$ plotted on the intervals $r \in [1, 4]$.
• Figure 3: The maximum eigenvalues of $r^2 \mathcal{P}^J(r)$ and $r^2 \tilde{\mathcal{P}}^J(r)$ (as functions of $r$) for $J \in \{0,1,2,3,4,5\}$ plotted on the intervals $r \in [1, 4]$.
• Figure 4: The maximum eigenvalue of $r^2 \mathcal{P}^0(r),r^2 \tilde{\mathcal{P}}^0(r)$, and $r^2 \mathcal{Q}^1(r)$ (as a function of $r$) plotted on the intervals $r \in [1, 1.5]$.
• Figure 5: $J = 0$: The maximum eigenvalues (as functions of $r$) of the 9 blocks in the further block decomposition $r^2\mathcal{R}^0(r)$ plotted on the interval $r \in [1, 4]$. The block $\mathcal{Q}^0_{10, 0}$ is the block with the positive eigenvalue.

Theorems & Definitions (179)

• Theorem 1
• Remark 2
• Corollary 3
• Corollary 4
• Remark 5
• Lemma 6
• proof
• Lemma 7
• proof
• Definition 8: Linear stability of a shrinking Ricci soliton