Linear stability and instability of Kähler Ricci solitons

TL;DR

The authors develop a weighted Weitzenböck framework tailored to 4-dimensional Kähler Ricci solitons and use it to analyze linear stability via the weighted Lichnerowicz Laplacian $L_f$. They show that all steady and expanding Kahler solitons are $L^2_f$-stable in this setting, with explicit decompositions into $J$-invariant and $J$-anti-invariant components, and they derive sharp bounds for these components. They demonstrate the shrinking BCCD soliton is linearly unstable by producing at least two independent weighted-harmonic $(1,1)$-forms, implying a positive second variation direction. Finally, they classify orbifold singularities: shrinking orbifold solitons are unstable, steady orbifolds neutral, and expanding orbifolds strictly stable, linking stability to the spectrum of the weighted self-dual curvature and soliton data.

Abstract

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of \cite{chi04} and \cite{hm11}, via recent work the \cite{cm21} on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted $L^2$-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

Theorems & Definitions (31)

• Theorem 1: Instability of the BCCD soliton
• Remark 2
• Theorem 3: Stability of steady and expanding Kähler–Ricci solitons
• Remark 4
• Theorem 5
• Remark 6
• Conjecture 7
• Remark 9
• Remark 10
• proof : Proof of Theorem \ref{['thm: stab']}