Linear stability and instability of K\"ahler Ricci solitons

TL;DR

This paper investigates the linear stability of Kähler Ricci solitons, showing instability of certain shrinking solitons and stability properties of steady and expanding types through spectral analysis and new Weitzenböck formulae.

Contribution

It introduces new Weitzenböck formulae for the weighted Lichnerowicz Laplacian and applies spectral analysis to determine stability of various Kähler Ricci solitons and their orbifold singularities.

Findings

BCCD shrinking soliton is linearly unstable.

Weighted L^2-spectra are nonpositive for steady and expanding solitons.

Shrinkers are unstable, steadies are neutrally stable, expanders are stable.

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 K\"ahler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of K\"ahler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenb\"ock formulae for the weighted Lichnerowicz Laplacian specialized to K\"ahler metrics.