On stability of the hyperbolic space form under the normalized Ricci flow (Q2785992)

The Ricci flow of Hamilton evolves the metric of a Riemannian manifold in the direction of an Einstein metric. \textit{R.-G. Ye} [``Ricci flow, Einstein metrics and space forms'', Trans. Am. Math. Soc. 338, No. 2, 871--896 (1993; Zbl 0804.53054)] constructed Einstein metrics on a compact Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) through the Ricci flow under an ``almost Einstein'' or Ricci pinching condition. A natural question is to consider a similar problem in non-closed Riemannian manifolds. NEWLINENEWLINEThis paper studies the normalized Ricci flow from a slight perturbation of the hyperbolic metric on \(\mathbb H^n\). Denote by \(g_{H}\) the hyperbolic metric with constant sectional curvature \(-1\) in the hyperbolic space \(\mathbb H^n\). A metric \(g\) on \(\mathbb H^n\) is said to be \(\varepsilon\)-hyperbolic for some positive \(\varepsilon>0\) ifNEWLINENEWLINE\[NEWLINE(1-\varepsilon)g_{H} \leq g\leq (1+\varepsilon)g_{ H}\, ,NEWLINE\]NEWLINENEWLINEandNEWLINENEWLINE\[NEWLINE\left| K(x,\sigma)+1\right|\leq \varepsilon,NEWLINE\]NEWLINENEWLINEwhere \(K(x,\sigma)\) is the sectional curvature of a tangent plane \(\sigma\) at \(x\in \mathbb H^n\). Given \(\delta>0\), the metric \(g\) is said to be \(\varepsilon\)-hyperbolic of order \(\delta\) if it additionally satisfiesNEWLINENEWLINE\[NEWLINE\left| (K(x,\sigma)+1)e^{\delta d(x,x_0)}\right|\leq \varepsilon,NEWLINE\]NEWLINENEWLINEwhere \(d(x, x_0)\) is the distance from \(x\) to some fixed point \(x_0\) with respect to the metric \(g.\) NEWLINENEWLINEThe first result proved by the authors is: For each \(n\geq 3\) and \(\delta>0\), there exists some \(\varepsilon>0\) depending only on \(\delta\) and \(n\) such that the normalized Ricci flow starting from any \(\varepsilon\)-hyperbolic metric \(g\) of order \(\delta\) on \(\mathbb H^n\) exists for all time and converges exponentially fast to some Einstein metric. NEWLINENEWLINEThe basic idea in the proof follows Ye's paper cited above, but instead of using an \(L^2\) estimate of the traceless Ricci tensor, they use a pointwise estimate of \(\left| R_{ij}+(n-1)g_{ij}\right| \) which allows them to handle the case of very small decay. The local estimate is used also to prove that, when the initial metric is asymptotically hyperbolic, the limiting Einstein metric is asymptotically hyperbolic of a certain degree, what combined with a rigidity result of \textit{Y-G. Shi} and \textit{G. Tian} [Commun. Math. Phys. 259, No.~3, 545--559 (2005; Zbl 1092.53033)], leads them to the second main result: For \(n>5\) and \(\delta> 2\), there exists \(\varepsilon>0\) depending only on \(\delta\) and \(n\) such that the normalized Ricci flow starting from any \(\varepsilon\)-hyperbolic metric of order \(\delta\) on \(\mathbb H^n\) converges exponentially fast to \(g_{H}\). NEWLINENEWLINEThe authors are not able to use their method to obtain the decay result for \(n\leq 5\) (although for \(n=3\) the result is still true, since every Einstein metric is of constant sectional curvature). However, their method does not work for \(n=4\), so \(n=5\) is the critical case with respect to the validity of the proof. They also show that the condition \(\delta>2\) is essential in the statement.