Linear stability of compact shrinking Ricci solitons

In this paper, we continue investigating the second variation of Perelman’s \(\nu \) $\nu$ -entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in Cao and Zhu (Math Ann 353(3):747–763, 2012), as well as the more recent work in Mehrmohamadi and Razavi (arXiv:2104.08343, 2021), and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in Cao et al. (arXiv:math.DG/0404165, 2004) (see also Cao and He in J Reine Angew Math, 2015:229–246, 2015) for positive Einstein manifolds to the compact shrinking Ricci soliton case.