Linear stability of compact shrinking Ricci solitons (Q6624805)

This paper concerns compact shrinking gradient Ricci solitons, which are closed Riemannian manifolds \((M,g)\) with Ricci tensor satisfying\N\[\N\mathrm{Ric}(g)+\nabla^{2}f=\frac{1}{2\tau}g,\N\]\Nwhere \(f\in C^{\infty}(M)\) and \(\tau>0\) is a constant. Ricci solitons can be considered generalisations of Einstein metrics (recovered by taking \(f\) constant) and appear naturally in the theory of the Ricci flow as both fixed points of the flow (up to diffeomorphism and homothetic scaling) and also as the geometries of certain singularity models.\N\NIn one of his breakthrough papers [``The entropy formula for the Ricci flow and its geometric applications'', Preprint, \url{arXiv:math/0211159} (2002; Zbl 1130.53001)], \textit{G. Perelman} discovered a function \(\nu\) on the set of Riemannian metrics that is increasing under the Ricci flow and has gradient shrinking Ricci solitons as its critical points. One can use \(\nu\) to investigate the behaviour of the Ricci flow at a particular soliton; if one can perturb the soliton in such a way that the entropy increases, then the fact that \(\nu\) is monotonically increasing means that the flow never returns to the soliton and hence it can be considered an unstable fixed point. The second variation of \(\nu\) was calculated by \textit{H.-D. Cao} et al. [``Gaussian densities and stability for some Ricci solitons'', Preprint, \url{arXiv:math/0404165}] and \textit{H.-D. Cao} and \textit{M. Zhu} [Math. Ann. 353, No. 3, 747--763 (2012; Zbl 1252.53074)]; a soliton is called linearly stable if the second variation of \(\nu\) is nonpositive. There is a general expectation that linearly stable (and more especially dynamically stable) solitons should be rather special; in particular, it might be possible to classify the stable shrinking Ricci solitons.\N\NWhen dealing with gradient Ricci solitons, it is useful to consider the \(L^{2}\)-inner product induced by the metric \(g\) but using the weighted volume form \({e^{-f}dV_{g}}\). This gives rise to certain differential operators that are `twisted' by the function \(f\); for example, there are the twisted divergence and Laplacian where, if \(h\in\mathrm{Sym}^{2}(T^{\ast}M)\),\N\[\N\mathrm{div}_{f}(h) = \mathrm{div}(h)-h(\nabla f,\cdot) \qquad \mathrm{and} \qquad \Delta_{f}(h) = \Delta h-\nabla_{\nabla f}h.\N\]\NAssociated to the second variation of Perelman's \(\nu\)-entropy is the following operator\N\[\N\mathcal{L}_{f}h:=\frac{1}{2}\Delta_{f}h+Rm(h,\cdot).\N\]\NThe authors consider the splitting\N\[\N\mathrm{Sym}^{2}(T^{\ast}M) = \mathrm{Im}(\mathrm{div}_{f}^{\dagger}) \oplus \mathbb{R}\cdot \mathrm{Ric} \oplus \mathrm{Ker}(\mathrm{div}_{f})_{0},\N\]\Nwhere \(\mathrm{div}_{f}^{\dagger}\) is the adjoint of \(\mathrm{div}_{f}\) and \(\mathrm{Ker}(\mathrm{div}_{f})_{0}\) are the tensors that are both \(\mathrm{div}_{f}\)-free and \(L^{2}\)-orthogonal to the Ricci tensor \(\mathrm{Ric}\) (in [\textit{H.-D. Cao} and \textit{M. Zhu}, Math. Ann. 353, No. 3, 747--763 (2012; Zbl 1252.53074)], the authors proved that the Ricci tensor satisfies \(\mathrm{div}_{f}(\mathrm{Ric})=0\)). The authors prove that the splitting is invariant with respect to the operator \(\mathcal{L}_{f}\) and that the eigenvalues of \(\mathcal{L}_{f}\) on the component \(\mathrm{Im}(\mathrm{div}_{f}^{\dagger})\) are less than \(\frac{1}{4\tau}\). They prove that a soliton is linearly stable if and only if \(\mathcal{L}_{f}\leq 0\) on the component \(\mathrm{Ker}(\mathrm{div}_{f})_{0}\).\N\NThe results in the paper under review build on those obtained by \textit{M. Mehrmohamadi} and \textit{A. Razavi} [Bull. Iran. Math. Soc. 50, No. 4, Paper No. 60, 26 p. (2024; Zbl 07950908)], who proved that the second variation of \(\nu\) vanishes on the \( \mathrm{Im}(\mathrm{div}_{f}^{\dagger})\) component of the splitting as well as providing lots of useful identities for various commutators involving the twisted differential operators mentioned previously. The authors of the paper under review develop and manipulate these identities in order to prove the main results. To the best of the reviewer's knowledge, there are no known examples of compact, non-Einstein, linearly stable shrinking solitons. The criterion for stability involving the eigenvalues of \(\mathcal{L}_{f}\) might be possible to be check on classes of soliton that have extra structure (e.g. Kähler-Ricci solitons), or even used to prove more general stability results; for example, the paper ends by mentioning a conjecture of the first author that in dimension four, a compact stable shrinking soliton must in fact be Einstein.