No local collapsing theorem I

In this section we present an application of the monotonicity formula (3.4) to the analysis of singularities of the Ricci flow.

4.1   Let $g_{ij}(t)$ be a smooth solution to the Ricci flow $(g_{ij})_t=-2R_{ij}$ on $[0,T).$ We say that $g_{ij}(t)$ is locally collapsing at $T,$ if there is a sequence of times $t_k\to T$ and a sequence of metric balls $B_k=B(p_k,r_k)$ at times $t_k,$ such that $r_k^2/t_k$ is bounded, $\vert Rm\vert(g_{ij}(t_k))\le r_k^{-2}$ in $B_k$ and $r_k^{-n}Vol(B_k)\to 0.$

Theorem. If $M$ is closed and $T<\infty,$ then $g_{ij}(t)$ is not locally collapsing at $T.$

Proof. Assume that there is a sequence of collapsing balls $B_k=B(p_k,r_k)$ at times $t_k\to T.$ Then we claim that $\mu(g_{ij}(t_k),r_k^2)\to -\infty.$ Indeed one can take $f_k(x)=-\log\phi($dist$_{t_k}(x,p_k)r_k^{-1})+c_k,$ where $\phi$ is a function of one variable, equal 1 on $[0,1/2],$ decreasing on $[1/2,1],$ and very close to 0 on $[1,\infty),$ and $c_k$ is a constant; clearly $c_k\to -\infty$ as $r_k^{-n}Vol(B_k)\to 0.$ Therefore, applying the monotonicity formula (3.4), we get $\mu(g_{ij}(0),t_k+r_k^2)\to -\infty.$ However this is impossible, since $t_k+r_k^2$ is bounded.

4.2   Definition We say that a metric $g_{ij}$ is $\kappa$-noncollapsed on the scale $\rho,$ if every metric ball $B$ of radius $r<\rho,$ which satisfies $\vert Rm\vert(x)\le r^{-2}$ for every $x\in B,$ has volume at least $\kappa r^n.$

It is clear that a limit of $\kappa$-noncollapsed metrics on the scale $\rho$ is also $\kappa$-noncollapsed on the scale $\rho;$ it is also clear that $\alpha^2g_{ij}$ is $\kappa$-noncollapsed on the scale $\alpha\rho$ whenever $g_{ij}$ is $\kappa$-noncollapsed on the scale $\rho.$ The theorem above essentially says that given a metric $g_{ij}$ on a closed manifold $M$ and $T<\infty,$ one can find $\kappa=\kappa(g_{ij},T)>0,$ such that the solution $g_{ij}(t)$ to the Ricci flow starting at $g_{ij}$ is $\kappa$-noncollapsed on the scale $T^{1/2}$ for all $t\in [0,T),$ provided it exists on this interval. Therefore, using the convergence theorem of Hamilton, we obtain the following

Corollary. Let $g_{ij}(t), t\in [0,T)$ be a solution to the Ricci flow on a closed manifold $M,$ $T<\infty.$ Assume that for some sequences $t_k\to T, p_k\in M$ and some constant $C$ we have $Q_k=\vert Rm\vert(p_k,t_k)\to\infty$ and $\vert Rm\vert(x,t)\le CQ_k,$ whenever $t<t_k.$ Then (a subsequence of) the scalings of $g_{ij}(t_k)$ at $p_k$ with factors $Q_k$ converges to a complete ancient solution to the Ricci flow, which is $\kappa$-noncollapsed on all scales for some $\kappa>0.$