Destabilising compact warped product Einstein manifolds (Q2063727)

In this well-written article the authors discuss the linear stability of warped product Einstein metrics as fixed points of the Ricci flow. The main motivation is that stable fixed points of the Ricci flow could be related to the proof of the Thurston's geometrization conjecture in large dimension. Recalling that Ricci solitons, in particular Einstein metrics, are fixed points of the Ricci flow, one can say that ``stability'' means that the Ricci flow starting at a small pertubation of a Ricci soliton will return to the soliton. The authors discuss the stability problem for the class of compact Einstein metrics \(g\) on a product manifold \(M=B\times F\), with \(g=\pi^*_B\bar{g}+(f\circ {\pi}_{B})^2\pi^*_F\tilde{g},\) where \(\pi_B,\pi_F\) are the natural projections, \(\bar{g},\tilde{g}\) Riemannian metrics on \(B\), \(F\) and \(f\in C^\infty(B)\), \(f\) never vanishing. The manifolds \(B,F\) are named the base and the fibre of \(M\). Several unstability results are obtained. In particular, we mention the following one. Theorem. Let \((M^n,g)\) be a warped product Einstein manifold. If \(n \leq 6\) and \(M\) is compact, then \((M,g)\) is an unstable fixed point of the Ricci flow. If \((M,g)\) is a warped product Einstein manifold whose fibre is an Einstein manifold, the authors formulate a condition on the eigentensors of the Lichnerowicz Laplacian of the fibre that implies the unstability of \((M,g)\). Another unstability result is proved for warped product Einstein manifolds which are constructed from a sequence of quasi-Einstein metrics on the base manifold that converges in the \(C^\infty\)-topology to a Ricci soliton. Finally, the authors state the unstability of suitable warped product metrics that are generalized Schwarzschild-Tangherini black holes.