Compact blow-up limits of finite time singularities of Ricci flow are shrinking Ricci solitons (Q2464261)

The Ricci flow equation on a closed differentiable manifold \(M\) is \[ \frac{\partial g( t) }{\partial t}=-2\operatorname{Ric} (g( t)), \] where \(\{ g( t) \mid t\in [ 0,T)\} \) is a family of Riemannian metrics on \(M\). A Ricci solution of the Ricci flow equation is given by \(g( t) =\alpha ( t) \Phi ^{\ast }( t) g_{0},\) where \(\alpha ( t) >0,\) \(\{ \Phi ( t) \mid t\in [ 0,T)\} \) is a family of diffeomorphisms of \(M\) and \(g_{0}\) is a fixed Riemannian metric on \(M\). A Ricci soliton \(g( t) =\alpha ( t) \Phi ^{\ast }( t) g_{0}\) is said to be shrinking if \( \alpha ^{\prime }( t) <0\). The author employs the \(\lambda \) and \( \mu \) functionals introduced by G. Perelman to prove his main result: if \( g( t)\), \(t\in [0,T)\), is a maximal solution of the Ricci flow equation with singular time \(T<+\infty \), that is, for subsequences \( t_{k}\to +\infty \) and \(\{ P_{k}\} _{k=1,2,\dots }\subset M\), the sequence \(\{ Q_{k}=| R_{m}| ( P_{k},t_{k}) \} _{k=1,2,\dots }\) converges to infinity, and if \( g( t) \) is such that the rescaled sequence \(\{ \langle M,Q_{k}g( Q_{k}^{-1}t+t_{k}) \rangle \} _{k=1,2,\dots }\) \(C^{\infty }\)-converges to a solution \(g_{\infty }( t) \), then \( g_{\infty }( t) \) must be a shrinking Ricci soliton.