  In this paper, we continue to investigate the second variation of Perelman's
$\nu$-entropy for compact shrinking Ricci solitons. In particular, we improve
some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No.
3, 747-763" and the more recent work in "M. Mansour and R. Asadollah,
arXiv:2104.08343" 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 "arXiv:math.DG/0404165"
(see also "H.-D. Cao and C. He, J. Reine Angew. Math. 2015 (2015), no. 709,
229-246.") for positive Einstein manifolds to the compact shrinking Ricci
soliton case.

Differential Geometry