[proofplan] The rescaling is chosen so that the selected curvature value becomes exactly $1$ at the basepoint and at rescaled time $0$. The lower endpoints of the rescaled time intervals tend to $-\infty$, while the forward endpoints have limiting availability governed precisely by $\liminf_j\lambda_j(T-t_j)$. The assumed local curvature bounds and basepoint noncollapsing give the hypotheses of pointed Hamilton compactness for Ricci flow, so a diagonal subsequence converges on all compact pointed parabolic regions. Finally, smooth convergence transfers the curvature normalization to the limit. [/proofplan]

[step:Verify that each rescaled metric solves Ricci flow on its rescaled time interval] Fix $j\in\mathbb{N}$. For $s\in I_j$, define the affine time-change map $\tau_j: I_j \to [0,T)$ by $\tau_j(s)=t_j+s/\lambda_j$. The definition of $I_j$ is exactly the condition $0<\tau_j(s)<T$, so $g_j(s)=\lambda_j g(\tau_j(s))$ is defined for every $s\in I_j$. Since $\lambda_j$ is constant in $s$, and since Ricci curvature is invariant under constant scaling of the metric in the sense that \begin{align*} \operatorname{Ric}_{\lambda_j g(\tau_j(s))}=\operatorname{Ric}_{g(\tau_j(s))}, \end{align*} we compute \begin{align*} \partial_s g_j(s)=\lambda_j\,\partial_t g(\tau_j(s))\,\frac{d\tau_j}{ds}(s)=\lambda_j\left(-2\operatorname{Ric}_{g(\tau_j(s))}\right)\frac{1}{\lambda_j}=-2\operatorname{Ric}_{g_j(s)}. \end{align*} Thus each $g_j: I_j \to \Gamma(S^2T^*M)$ is a Ricci flow. [/step]

[step:Normalize the curvature at the selected basepoint]For a constant metric scaling $\tilde g=\lambda g$ with $\lambda>0$, the curvature norm scales by \begin{align*} |\operatorname{Rm}_{\tilde g}|_{\tilde g}=\lambda^{-1}|\operatorname{Rm}_{g}|_{g}. \end{align*} Applying this with $\tilde g=g_j(0)=\lambda_j g(t_j)$ gives \begin{align*} |\operatorname{Rm}_{g_j(0)}|(x_j)=\lambda_j^{-1}|\operatorname{Rm}_{g(t_j)}|(x_j)=\lambda_j^{-1}\lambda_j=1. \end{align*}[/step]

[guided]The point of the factor $\lambda_j$ is to measure geometry in the curvature scale at $(x_j,t_j)$. At rescaled time $s=0$, the original time is \begin{align*} \tau_j(0)=t_j+\frac{0}{\lambda_j}=t_j, \end{align*} so \begin{align*} g_j(0)=\lambda_j g(t_j). \end{align*} Under the constant scaling $\tilde g=\lambda g$, distances scale by $\lambda^{1/2}$ and curvature norms scale by $\lambda^{-1}$. Therefore, with $\lambda=\lambda_j$, \begin{align*} |\operatorname{Rm}_{g_j(0)}|(x_j)=|\operatorname{Rm}_{\lambda_j g(t_j)}|(x_j)=\lambda_j^{-1}|\operatorname{Rm}_{g(t_j)}|(x_j). \end{align*} By definition of the blow-up factor, \begin{align*} \lambda_j=|\operatorname{Rm}_{g(t_j)}|(x_j), \end{align*} and hence \begin{align*} |\operatorname{Rm}_{g_j(0)}|(x_j)=1. \end{align*} This is the normalization that survives in the limiting flow.[/guided]

[step:Show that every compact backward time interval eventually lies in the rescaled domains] Let $A<0$ be fixed. Since $t_j\uparrow T$ and $T>0$, there exists $j_1\in\mathbb{N}$ such that \begin{align*} t_j\ge \frac{T}{2} \end{align*} for all $j\ge j_1$. Since $\lambda_j\to\infty$, there exists $j_2\in\mathbb{N}$ such that \begin{align*} \lambda_j\frac{T}{2}>-A \end{align*} for all $j\ge j_2$. Hence, for $j\ge \max\{j_1,j_2\}$, \begin{align*} -\lambda_j t_j\le -\lambda_j\frac{T}{2}<A. \end{align*} Because $0<\lambda_j(T-t_j)$, this proves \begin{align*} [A,0]\subset I_j \end{align*} for all sufficiently large $j$. [/step]

[step:Apply pointed compactness on each backward interval and diagonalize]Fix $A<0$ and $R<\infty$. Let $\iota_0>0$ denote the uniform basepoint injectivity-radius constant from the basepoint compactness noncollapsing hypothesis in the statement. By hypothesis, the curvatures of $g_j(s)$ are uniformly bounded on every compact pointed parabolic neighborhood contained in the domains $I_j$; in particular, for all sufficiently large $j$ they are uniformly bounded on \begin{align*} B_{g_j(0)}(x_j,R)\times[A,0]. \end{align*} The same basepoint compactness noncollapsing hypothesis gives \begin{align*} \operatorname{inj}_{g_j(0)}(x_j)\ge \iota_0 \end{align*} for all sufficiently large $j$. Thus the hypotheses of Hamilton's pointed compactness theorem for Ricci flow are satisfied on $[A,0]$: the approximating flows are complete because $M$ is compact, the basepoint injectivity radii have the positive lower bound $\iota_0$, and the curvature is uniformly bounded on each compact pointed parabolic neighborhood. The required higher derivative bounds on smaller compact parabolic neighborhoods are supplied by [Shi local derivative estimates](/theorems/5983) applied inside those bounded-curvature neighborhoods. Hence, for each fixed $A<0$, Hamilton compactness gives a subsequence converging smoothly in the pointed Cheeger-Gromov sense on compact subsets of space-time over $[A,0]$ to a complete pointed Ricci flow; completeness of the limit is part of this compactness conclusion under the complete approximating flows and the basepoint injectivity-radius lower bound. Applying this argument to the nested intervals \begin{align*} [-1,0],\ [-2,0],\ [-3,0],\dots \end{align*} and taking a diagonal subsequence gives a single pointed smooth limit $ (M_\infty,g_\infty(s),x_\infty)$ for $s\in(-\infty,0]$.[/step]

[guided]We now verify the [compactness theorem](/theorems/2748) rather than treating compactness as automatic. Fix $A<0$ and $R<\infty$. Since the previous step showed that $[A,0]\subset I_j$ for all sufficiently large $j$, the parabolic region \begin{align*} B_{g_j(0)}(x_j,R)\times[A,0] \end{align*} is contained in the time domain of $g_j$ after discarding finitely many indices. The curvature-boundedness assumption in the theorem statement applies to every compact pointed parabolic neighborhood contained in $I_j$, so the curvature tensors $\operatorname{Rm}_{g_j(s)}$ are uniformly bounded on this region. Hamilton's pointed compactness theorem for Ricci flow also needs a basepoint injectivity-radius lower bound at one time slice. The basepoint compactness noncollapsing condition in the statement is precisely this condition: there exists a constant $\iota_0>0$ such that \begin{align*} \operatorname{inj}_{g_j(0)}(x_j)\ge \iota_0 \end{align*} for all sufficiently large $j$. This is the point where the noncollapsing hypothesis is used. Without this positive lower bound, bounded curvature alone would not prevent pointed collapse at the basepoints, and Cheeger-Gromov convergence to a manifold limit could fail. The approximating flows are complete because the original manifold $M$ is compact and constant rescaling preserves completeness. The local curvature bounds give the zeroth-order geometric control required by Hamilton's pointed compactness theorem for Ricci flow, while Shi local derivative estimates convert those curvature bounds into uniform bounds for all covariant derivatives of curvature on slightly smaller compact parabolic neighborhoods. Therefore the compactness theorem applies on $[A,0]$ and produces a subsequence converging smoothly in the pointed Cheeger-Gromov sense on compact subsets of space-time to a complete pointed Ricci flow.[/guided]

[step:Pass the curvature normalization to the smooth limit] Pointed smooth Cheeger-Gromov convergence means that, after choosing an exhaustion of $M_\infty$ by relatively compact open sets and smooth embeddings into $M$, the pulled-back metrics converge in $C^\infty$ on compact subsets of space-time. In particular, the pulled-back curvature tensors converge smoothly, and therefore their pointwise norms converge at the basepoint and time $0$. Since $|\operatorname{Rm}_{g_j(0)}|(x_j)=1$ for every $j$, the limit satisfies \begin{align*} |\operatorname{Rm}_{g_\infty(0)}|(x_\infty)=1. \end{align*} [/step]

[step:Extend the same subsequential limit forward up to the liminf time] Assume \begin{align*} \omega=\liminf_{j\to\infty}\lambda_j(T-t_j)>0. \end{align*} Let $B$ be any real number with $B<\omega$. By the definition of the liminf, there exist $\varepsilon>0$ and $j_3\in\mathbb{N}$ such that \begin{align*} B+\varepsilon<\omega \end{align*} and \begin{align*} \lambda_j(T-t_j)>B \end{align*} for all $j\ge j_3$. Combining this with the backward endpoint argument, for every $A<0$ we have \begin{align*} [A,B]\subset I_j \end{align*} for all sufficiently large $j$. Because the curvature-bound hypothesis in the statement applies to every compact pointed parabolic neighborhood contained in the rescaled domains, and because the basepoint injectivity-radius lower bound $\operatorname{inj}_{g_j(0)}(x_j)\ge \iota_0$ is unchanged, the same hypotheses of Hamilton's pointed compactness theorem for Ricci flow hold on each interval $[A,B]$ with $A<0<B<\omega$ after discarding finitely many indices. Therefore Hamilton compactness applies on each such compact time interval, with higher derivative bounds again supplied by Shi local derivative estimates. Diagonalizing over a countable exhaustion of $(-\infty,\omega)$ by compact intervals yields a complete pointed Ricci flow $(M_\infty,g_\infty(s),x_\infty)$ for $s\in(-\infty,\omega)$. Its restriction to $(-\infty,0]$ is the backward limit already constructed, because the diagonal subsequence is chosen to refine the preceding subsequence on every compact interval containing $0$. This proves the claimed forward extension. [/step]