The second variation of the Ricci expander entropy (Q635683)

If \((M^{n},g)\) is a Riemannian manifold with metric \(g\), as a generalization of an Einstein metric, \(g\) is a Ricci soliton if \(\frac{1}{2}\mathfrak{L} _{X}g+Ric-\lambda g=0\) where \(\mathfrak{L}_{X}\) is the Lie derivative in the direction of a vector field \(X\), \(Ric\) is the Ricci curvature of \(M\) and \( \lambda \in R\) is a constant on \(M.\) A soliton is said to be expanding for \( \lambda <0\), steady for \(\lambda \) \(=0\) and contracting for \(\lambda >0.\) The paper deals with soliton solutions of the Ricci flow \(\frac{\partial g_{ij}}{\partial t}\) \(=-2R_{ij}\). The author computes the first and second variations of the entropy functional \(v_{+}\) of the functional \(W_{+}\) introduced by \textit{M. Feldman}, \textit{T. Ilmanen} and \textit{L. Ni} [J. Geom. Anal. No. 1, 49--62 (2005; Zbl 1071.53040)] for the case of expanding Ricci solitons where \(v_{+} \) is nondecreasing along the Ricci flow and constant on expanding solitons. From the computation of the first variation of \(v_{+}\) the critical points of \(W_{+}\) are the expanding Ricci solitons which on compact Riemannian manifolds coincide with negative Einstein metrics. As an application the author considers the linear stability of negative Einstein manifolds. He shows that a compact real hyperbolic space form in dimension \(n\geq 3\) is linearly stable.