Almost-rigidity and the extinction time of positively curved Ricci flows (Q2405962)

The paper contains three main results. In the first result the authors show that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with \(\mathbb{R}^2\). Theorem 1. Let \((M^n,g_0)\), \(n\geq 3\), be a Riemannian manifold such that its scalar curvature \(R_{g_0}\geq n(n-1)\) and \((M^n,g_0)\times\mathbb{R}^{2}\) has curvature in the interior of the cone of positive isotropic curvature operators. Given \(\eta>0\), there exists a number \(\tau>0\), which only depends on \(\eta\) and on the topology of \(M\), such that if the Ricci flow evolution \(g(t)\) of \(g_0\) has singular time \(T>\frac{1}{2(n-1)}-\tau\), then \(g_0\) is \(\eta\)-close to a metric of constant curvature 1 in the \(C^0\)-norm. As an application of this result, the authors show that positively curved metrics on \(S^3\) and \(\mathbb{R}P^3\) with almost maximal width must be nearly round. Theorem 2. Let \(g\) be a Riemannian metric on the 3-sphere \(S^3\) with positive sectional curvature and scalar curvature \(R\geq6\). Given \(\eta>0\), there exists \(\varepsilon=\varepsilon(\eta)>0\) such that if the width of \(g\) satisfies \(W(g)>4\pi-\varepsilon\), then \(g\) is \(\eta\)-close to a metric of constant curvature 1 in the \(C^0\)-norm. Theorem 3. Let \(g\) be a Riemannian metric on real projective 3-space \(\mathbb{R}P^3\) with positive sectional curvature and scalar curvature \(R\geq 6\). Given \(\eta>0\), there exists \(\varepsilon=\varepsilon(\eta)>0\) such that if the least-area embedding of \(\mathbb{R}P^2\) satisfies \(\mathcal{A}(g)>2\pi-\varepsilon\), then \(g\) is \(\eta\)-close to a metric of constant curvature 1 in the \(C^0\)-norm.