On the linear stability of K\"ahler-Ricci solitons

TL;DR

This paper proves that K"ahler-Ricci solitons with a certain cohomology dimension are linearly unstable, extending previous results from the K"ahler-Einstein case to a broader class.

Contribution

It establishes the linear instability of K"ahler-Ricci solitons with high cohomology dimension, generalizing prior stability results.

Findings

K"ahler-Ricci solitons with dim H^{(1,1)}(M) ≥ 2 are linearly unstable

Extension of Cao-Hamilton-Ilmanen's results from K"ahler-Einstein to K"ahler-Ricci solitons

Provides new insights into the stability properties of K"ahler-Ricci solitons

Abstract

This is a short note proving that K\"ahler-Ricci solitons with $\dim H^{(1,1)}(M)\ge 2$ are linearly unstable. This extends the results of Cao-Hamilton-Ilmanen in the K\"ahler-Einstein case.