Linear and dynamical stability of Ricci-flat metrics

Abstract

We can talk about two kinds of stability of the Ricci flow at Ricci-flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$ (see [1, page 5]). The other one is a dynamical stability, and it refers to a convergence of a Ricci flow starting at any metric in a neighborhood of a considered Ricci-flat metric. We show that dynamical stability implies linear stability. We also show that a linear stability together with the integrability assumption implies dynamical stability. As a corollary, we get a stability result for $K3$-surfaces, part of which has been done in [11, Corollary 4.15, Theorem 4.16]. Our stability result applies to Calabi-Yau manifolds as well