A general convergence result for the Ricci flow in higher dimensions (Q959622)

Let \((M,g_0)\) be a compact Riemannian manifold of dimension \(n\geq 4.\) In this paper, the author studies the long-time behavior of solutions of the normalized Ricci flow, that is, one-parameter families of metrics \(g(t)\) such that \[ {\partial\over{\partial t}}g(t)=-2\text{Ric}_{g(t)}+{2\over{n}}r_{g(t)}g(t), \] where \(\text{Ric}_{g(t)}\) is the Ricci tensor of \(g(t)\), and \(r_{g(t)}\) is the mean value of the scalar curvature of \(g(t).\) The main result he proves is that if \[ R_{1313}+\lambda^2 R_{1414}+R_{2323}+\lambda^2 R_{2424}-2\lambda R_{1234}>0, \] for all orthonormal \(4\)-frames \(\{e_1,\ldots,e_4\}\) and all \(\lambda\in[-1,1]\), that is, if \((M,g_0)\times{\mathbb R}\) has positive isotropic curvature, then the normalized Ricci flow with initial metric \(g_0\) exists for all times and converges to a constant curvature metric as \(t\to\infty.\)