[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*} [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] [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](/page/Hamilton%20Compactness%20Theorem) 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](/page/Shi%20Derivative%20Estimates) 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. [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](/page/Hamilton%20Compactness%20Theorem), while [Shi local derivative estimates](/page/Shi%20Derivative%20Estimates) 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] 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] [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](/page/Hamilton%20Compactness%20Theorem) 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](/page/Shi%20Derivative%20Estimates). 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]