Perelman's Ricci steady and shrinker entropies, lambda( g) and nu(g), and the Ricci expander entropy, nu +(g), introduced by Feldman-Ilmanen-Ni are nondecreasing along the Ricci flow and their critical points are exactly compact gradient steady (i.e., Ricci flat), shrinking and expanding (i.e., negative Einstein) Ricci solitons, respectively.;In [1], Cao-Hamilton-Ilmanen presented the second variations of lambda( g) and nu(g) and investigated the entropy stability of compact Ricci flat and positive Einstein manifolds. In this paper, we first compute the second variation of nu+(g) and briefly discuss the entropy stability of compact hyperbolic space forms. Next, we calculate the second variation of nu(g) for general compact gradient shrinking solitons which was essentially due to Cao-Hamilton-Ilmanen (first stated in [2], see also [3]). Our main contributions are that we give all the computational detail which Cao-Hamilton-Ilmanen did not show, and the last term in their formula was corrected. As an application of this formula, we obtain a necessary condition for entropy stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.;Finally, we study the rigidity of gradient Kahler-Ricci solitons with harmonic Bochner tensor. In particular, we prove that complete gradient steady Kahler-Ricci solitons with harmonic Bochner tensor are Kahler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (respectively, expanding) Kahler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of Nkx Cn-k , where N is a Kahler-Einstein manifold with positive (respectively, negative) scalar curvature.
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