Linear stability of Perelman's $ν$-entropy on symmetric spaces of compact type

Abstract: Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's $\nu$-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the $\nu$-entropy on symmetric spaces of compact type. In particular, we exhibit many more linearly stable and linearly unstable examples than previously known and also the first linearly stable examples, other than the standard spheres, whose second variations are negative definite.