On second variation of Perelman's Ricci shrinker entropy (Q431243)

The authors provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and Huai-Dong Cao (see Theorem 6.3 of [\textit{H.-D. Cao}, Advanced Lectures in Mathematics (ALM) 11, 1--38 (2010; Zbl 1201.53046)] for Perelman's \(\nu\)-entropy. In particular, they correct an error in the stability operator stated in Theorem 6.3 of [loc. cit.]. Moreover, the authors obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.