[proofplan] We first turn the stated curvature hypotheses into uniform local space-time bounds for all covariant derivatives of curvature, using Shi's local derivative estimates on compact time subintervals. The injectivity-radius lower bound at the basepoints and the curvature bound at time $0$ give uniform pointed coordinate control on larger and larger balls. Arzela-Ascoli compactness is then applied in these coordinates, and a diagonal subsequence produces a smooth pointed Cheeger-Gromov limit on every compact subset of space-time. Finally, the Ricci flow equation passes to the limit and completeness follows from pointed smooth convergence together with the uniform local metric control. [/proofplan]

[step:Choose an exhaustion of the pointed space-time regions] Choose compact intervals $J'_m := [\alpha_m,\beta_m] \subset (a,b)$ and $J_m := [a_m,b_m] \subset (a,b)$ such that $0 \in J'_m$, $J'_m \Subset J_m$, $J'_m \subset J'_{m+1}$, and $\bigcup_{m=1}^{\infty} J'_m = (a,b)$. We also choose an increasing sequence of radii $R_m \to \infty$ with $R_m \ge m+1$. For each $m \in \mathbb{N}$, let $B_{j,m}$ denote the $g_j(0)$-geodesic ball $B_{g_j(0)}(x_j,R_m) \subset M_j$. The strengthened curvature hypothesis supplies the constant $C_m := C(J_m)$ such that \begin{align*} |\operatorname{Rm}_{g_j(t)}|_{g_j(t)}(x) \le C_m \end{align*} for every $j \in \mathbb{N}$, every $t \in J_m$, and every $x \in B_{j,m}$. [/step]

[step:Promote curvature bounds to uniform local derivative and metric bounds]Fix $m \in \mathbb{N}$. First use only the zeroth-order curvature bound on $B_{j,m} \times J_m$. Since $|\operatorname{Ric}_{g_j(t)}|_{g_j(t)} \le n|\operatorname{Rm}_{g_j(t)}|_{g_j(t)} \le nC_m$ there, the Ricci flow equation $\partial_t g_j(t)=-2\operatorname{Ric}_{g_j(t)}$ and Gronwall's Inequality imply that, with $E_m = \exp(2nC_m|J_m|)$, \begin{align*} E_m^{-1} g_j(0) \le g_j(t) \le E_m g_j(0) \end{align*} on $B_{g_j(0)}(x_j,R_m)$ for every $t \in J_m$. Define the spatial margin $\rho_m := (2E_m^{1/2})^{-1}$. If $x \in B_{g_j(0)}(x_j,R_m-1)$ and $y \in B_{g_j(t)}(x,\rho_m)$ for some $t \in J'_m$, then the metric comparison gives $d_{g_j(0)}(x,y) \le E_m^{1/2}d_{g_j(t)}(x,y) < 1/2$, hence $y \in B_{g_j(0)}(x_j,R_m)$. Thus every evolving $g_j(t)$-ball $B_{g_j(t)}(x,\rho_m)$ needed below is contained in the fixed ball on which the curvature bound is known. Since each flow is complete and $J'_m$ has positive distance from $\partial J_m$, Shi's Local Derivative Estimates apply on these contained parabolic neighbourhoods. For every integer $q \ge 0$, they give a constant $A_{m,q}<\infty$, depending only on the dimension, $q$, $C_m$, $\rho_m$, and $\operatorname{dist}(J'_m,\partial J_m)$, such that \begin{align*} |\nabla_{g_j(t)}^q \operatorname{Rm}_{g_j(t)}|_{g_j(t)}(x) \le A_{m,q} \end{align*} whenever $t \in J'_m$ and $x \in B_{g_j(0)}(x_j,R_m-1)$. The dependence on $\rho_m$ and $\operatorname{dist}(J'_m,\partial J_m)$ is exactly the dependence coming from the spatial cutoff scale and the backward-and-forward time margin in Shi's parabolic local estimate.[/step]

[guided]The derivative estimate requires more than the formal statement that curvature is bounded on a fixed $g_j(0)$-ball. Shi's local derivative estimates are parabolic local estimates: to apply them at a point $(x,t)$, we must know that a definite evolving metric ball around $x$ stays inside the region where the curvature bound is available, and that the time $t$ stays away from the boundary of the time interval. We first obtain the needed comparison between $g_j(t)$ and $g_j(0)$ without using any derivative estimate. On $B_{g_j(0)}(x_j,R_m) \times J_m$, the curvature hypothesis gives \begin{align*} |\operatorname{Ric}_{g_j(t)}|_{g_j(t)} \le n|\operatorname{Rm}_{g_j(t)}|_{g_j(t)} \le nC_m. \end{align*} For every point $x \in B_{g_j(0)}(x_j,R_m)$ and tangent vector $v \in T_xM_j$, the Ricci flow equation gives \begin{align*} \left|\frac{d}{dt}g_j(t)(v,v)\right| = 2|\operatorname{Ric}_{g_j(t)}(v,v)| \le 2nC_m g_j(t)(v,v). \end{align*} Applying Gronwall's Inequality on $J_m$ yields \begin{align*} E_m^{-1}g_j(0)(v,v) \le g_j(t)(v,v) \le E_m g_j(0)(v,v). \end{align*} Here \begin{align*} E_m = \exp(2nC_m|J_m|). \end{align*} This verifies a fixed comparison constant independent of $j$. Now set $\rho_m := (2E_m^{1/2})^{-1}$. Suppose $x \in B_{g_j(0)}(x_j,R_m-1)$, $t \in J'_m$, and $y \in B_{g_j(t)}(x,\rho_m)$. The metric comparison implies \begin{align*} d_{g_j(0)}(x,y) \le E_m^{1/2}d_{g_j(t)}(x,y) < \frac{1}{2}. \end{align*} Therefore \begin{align*} d_{g_j(0)}(y,x_j) \le d_{g_j(0)}(y,x)+d_{g_j(0)}(x,x_j). \end{align*} The right-hand side is bounded by \begin{align*} d_{g_j(0)}(y,x)+d_{g_j(0)}(x,x_j) < \frac{1}{2}+R_m-1 < R_m, \end{align*} so $y \in B_{g_j(0)}(x_j,R_m)$. This is the containment that lets us feed the fixed-ball curvature hypothesis into the local parabolic estimate. The hypotheses of Shi's Local Derivative Estimates are now verified: the flows are complete by assumption, curvature is bounded by $C_m$ on the contained parabolic neighbourhoods, the spatial radius is bounded below by $\rho_m$, and the time interval $J'_m$ has positive distance from $\partial J_m$. Hence for every integer $q \ge 0$ there exists a constant $A_{m,q}<\infty$, depending only on the dimension, $q$, $C_m$, $\rho_m$, and $\operatorname{dist}(J'_m,\partial J_m)$, such that \begin{align*} |\nabla_{g_j(t)}^q \operatorname{Rm}_{g_j(t)}|_{g_j(t)}(x) \le A_{m,q} \end{align*} for all $j$, all $t \in J'_m$, and all $x \in B_{g_j(0)}(x_j,R_m-1)$. The constant records two geometric margins: the lower spatial radius $\rho_m$ available for the cutoff balls and the positive time separation $\operatorname{dist}(J'_m,\partial J_m)$ needed by the parabolic estimate.[/guided]

[step:Apply pointed Cheeger-Gromov compactness on each fixed scale] At time $0$, the bounds $|\operatorname{Rm}_{g_j(0)}| \le C_m$ on $B_{j,m}$ and $\operatorname{inj}_{g_j(0)}(x_j) \ge i_0$ give uniform noncollapsing first at the basepoint and then at all points in the smaller pointed ball. Define the positive radius $s_m$ by \begin{align*} s_m := \frac{1}{16}\min\{1,i_0,R_m^{-1}\}. \end{align*} The curvature bound gives a sectional-curvature bound $|\operatorname{sec}_{g_j(0)}|\le C_m$ on $B_{j,m}$. Since $s_m \le i_0/16$, the exponential map at $x_j$ is nonsingular on the $g_j(0)$-ball of radius $4s_m$ in $T_{x_j}M_j$. The Rauch comparison estimate in these [normal coordinates](/theorems/2713) gives a constant $v_m>0$, depending only on the dimension, $C_m$, and $i_0$, such that \begin{align*} \operatorname{Vol}_{g_j(0)} B_{g_j(0)}(x_j,s_m) \ge v_m. \end{align*} Now fix $y \in B_{g_j(0)}(x_j,R_m-1)$. Choose a minimizing $g_j(0)$-geodesic segment from $x_j$ to $y$, and choose points $z_0=x_j,z_1,\dots,z_N=y$ on that segment with $N\le \lceil 8R_m/s_m\rceil$ and $d_{g_j(0)}(z_{\ell-1},z_\ell)\le s_m/8$. For every $\ell$, the ball $B_{g_j(0)}(z_\ell,4s_m)$ is contained in $B_{g_j(0)}(x_j,R_m)$ because $d_{g_j(0)}(z_\ell,x_j)\le R_m-1$ and $4s_m\le 1/4$. On each such contained ball we have the Ricci lower bound $\operatorname{Ric}_{g_j(0)}\ge -(n-1)C_m g_j(0)$. Applying the local Bishop-Gromov comparison estimate on $B_{g_j(0)}(z_\ell,4s_m)$ gives a doubling constant $D_m<\infty$, depending only on the dimension and $C_m s_m^2$, such that lower volume of a ball of radius $s_m$ centered at $z_{\ell-1}$ implies lower volume of a ball of radius $s_m$ centered at $z_\ell$. Iterating along the chain gives \begin{align*} \operatorname{Vol}_{g_j(0)} B_{g_j(0)}(y,s_m) \ge D_m^{-N} v_m =: v'_m >0, \end{align*} where $v'_m$ depends only on the dimension, $R_m$, $C_m$, and $i_0$. The ball containment needed for the Cheeger-Gromov-Taylor Injectivity Radius Estimate is the same one: $B_{g_j(0)}(y,4s_m)\subset B_{g_j(0)}(x_j,R_m)$. Applying that estimate with sectional-curvature bound $C_m$, radius $s_m$, and volume lower bound $v'_m$ gives a number $\iota_m>0$, depending only on the dimension, $R_m$, $C_m$, and $i_0$, such that every point of $B_{g_j(0)}(x_j,R_m-1)$ has $g_j(0)$-injectivity radius at least $\iota_m$. Combining this injectivity-radius lower bound with the curvature and curvature-derivative bounds at time $0$ gives a uniform harmonic-coordinate radius on these smaller balls by the harmonic-coordinate compactness criterion for Riemannian metrics with uniformly bounded curvature derivatives and uniformly positive injectivity radius. In those harmonic charts, the estimates from the previous step give uniform $C^q$ bounds for the coefficient functions of $g_j(t)$, including their time derivatives after differentiating the Ricci flow equation, on compact subsets of coordinate domains for every $q \ge 0$ and every $t \in J'_m$. Therefore the hypotheses of Pointed Cheeger-Gromov Compactness are satisfied on the fixed scale $m$ in the following concrete sense: after passing to a subsequence, there is a pointed manifold-with-precompact-domain $(U_m,g_{\infty,m}(0),x_{\infty,m})$ and smooth embeddings \begin{align*} \Phi_{j,m}: U_m &\to M_j \end{align*} with $\Phi_{j,m}(x_{\infty,m})=x_j$ and with images containing the controlled pointed region, such that $\Phi_{j,m}^*g_j(t)$ converges in $C^\infty$ on compact subsets of $U_m \times J'_m$. The [Arzela-Ascoli Theorem](/theorems/66) is the compactness mechanism in each fixed harmonic coordinate chart, and the pointed Cheeger-Gromov theorem supplies the transition compatibility between overlapping charts. [/step]

[step:Extract one diagonal subsequence on all compact scales] For $m=1$, choose a subsequence with smooth convergence on the first pointed region and embeddings $\Phi_{j,1}:U_1\to M_j$. From that subsequence choose a further subsequence with smooth convergence on the second region and embeddings $\Phi_{j,2}:U_2\to M_j$, and continue inductively. The pointed Cheeger-Gromov construction may be arranged so that, after restricting $U_m$ slightly if necessary, the transition maps $\Phi_{j,m+1}^{-1}\circ \Phi_{j,m}$ converge smoothly on compact subsets of $U_m$ to smooth embeddings \begin{align*} \Psi_m: U_m &\to U_{m+1} \end{align*} which identify the limiting metrics on overlaps. Let $(j_\ell)$ be the diagonal subsequence, so that for every fixed $m$, the sequence indexed by $j_\ell$ eventually lies in the subsequence chosen at stage $m$. Define $M_\infty$ as the direct limit of the domains $U_m$ under the embeddings $\Psi_m$, and let $x_\infty$ be the common image of the basepoints $x_{\infty,m}$. The overlap identities imply that the local families $g_{\infty,m}(t)$ patch to a smooth family of metrics $g_\infty(t)$ for every $t \in (a,b)$. Because the intervals $J'_m$ exhaust $(a,b)$ and the radii $R_m-1$ tend to infinity, the convergence along the diagonal subsequence is smooth on every compact subset in the pointed Cheeger-Gromov sense. [/step]

[step:Pass the Ricci flow equation and completeness to the limit]The convergence is $C^\infty$ in space and time on compact subsets, so the tensor identity $\partial_t g_j(t)=-2\operatorname{Ric}_{g_j(t)}$ passes to the limit and gives \begin{align*} \partial_t g_\infty(t) = -2\operatorname{Ric}_{g_\infty(t)}. \end{align*} Thus $(M_\infty,g_\infty(t))$ is a Ricci flow on $(a,b)$. It remains to prove completeness. First consider time $0$. For each $m$, the pointed Cheeger-Gromov construction gives a compact limiting domain $K_m \subset M_\infty$ corresponding to the controlled region $B_{g_j(0)}(x_j,R_m-1)$, with $K_m \subset K_{m+1}$ and $\bigcup_m K_m=M_\infty$. Since the source manifolds $(M_j,g_j(0))$ are complete and the controlled radii $R_m-1$ tend to infinity, the construction gives \begin{align*} \operatorname{dist}_{g_\infty(0)}(x_\infty,\partial K_m) \to \infty. \end{align*} Thus every piecewise $C^1$ curve escaping every compact subset of $M_\infty$ has infinite $g_\infty(0)$-length, because it must leave each $K_m$ and hence must have length at least $\operatorname{dist}_{g_\infty(0)}(x_\infty,\partial K_m)$ for arbitrarily large $m$. By the Hopf-Rinow characterization of completeness for connected Riemannian manifolds, $g_\infty(0)$ is complete. Now fix $t_0 \in (a,b)$. Choose a compact interval $J\subset(a,b)$ containing both $0$ and $t_0$, and let $C_J:=C(J)$ be the global curvature bound from the theorem statement. The Ricci estimate $|\operatorname{Ric}_{g_j(t)}|_{g_j(t)}\le nC_J$ on $M_j\times J$, the Ricci flow equation, and Gronwall's Inequality give the global comparison \begin{align*} E_J^{-1}g_j(0) \le g_j(t_0) \le E_Jg_j(0), \qquad E_J:=\exp(2nC_J|J|), \end{align*} on all of $M_j$. Passing to the pointed smooth limit on each $K_m$ gives \begin{align*} E_J^{-1}g_\infty(0) \le g_\infty(t_0) \le E_Jg_\infty(0) \end{align*} on every $K_m$, hence on all of $M_\infty$. Therefore any escaping piecewise $C^1$ curve has $g_\infty(t_0)$-length at least $E_J^{-1/2}$ times its $g_\infty(0)$-length, which is infinite. By Hopf-Rinow again, $g_\infty(t_0)$ is complete. Since $t_0 \in (a,b)$ was arbitrary, every limiting metric is complete, and the diagonal subsequence converges smoothly in the pointed Cheeger-Gromov sense to the complete pointed Ricci flow $(M_\infty,g_\infty(t),x_\infty)$ on $(a,b)$.[/step]

[guided]The completeness argument is easiest after using the strengthened global curvature hypothesis. We first prove completeness at time $0$, then transfer it to an arbitrary time by a global bilipschitz comparison. For time $0$, the pointed Cheeger-Gromov construction gives compact limiting domains $K_m\subset M_\infty$ corresponding to the regions $B_{g_j(0)}(x_j,R_m-1)$, with $K_m\subset K_{m+1}$ and $\bigcup_m K_m=M_\infty$. Because the approximating manifolds $(M_j,g_j(0))$ are complete and the radii $R_m-1$ tend to infinity, these limiting domains recede from the basepoint: \begin{align*} \operatorname{dist}_{g_\infty(0)}(x_\infty,\partial K_m) \to \infty. \end{align*} If a piecewise $C^1$ curve escapes every compact subset of $M_\infty$, then it eventually leaves each $K_m$. Its $g_\infty(0)$-length is therefore at least $\operatorname{dist}_{g_\infty(0)}(x_\infty,\partial K_m)$ for arbitrarily large $m$, and hence is infinite. The Hopf-Rinow characterization of completeness for connected Riemannian manifolds then implies that $g_\infty(0)$ is complete. Now let $t_0\in(a,b)$. Choose a compact interval $J\subset(a,b)$ containing $0$ and $t_0$. The global curvature hypothesis gives a constant $C_J=C(J)$ with $|\operatorname{Rm}_{g_j(t)}|_{g_j(t)}\le C_J$ on all of $M_j\times J$. Hence \begin{align*} |\operatorname{Ric}_{g_j(t)}|_{g_j(t)}\le nC_J. \end{align*} For every $x\in M_j$ and $v\in T_xM_j$, the Ricci flow equation gives \begin{align*} \left|\frac{d}{dt}g_j(t)(v,v)\right|\le 2nC_J g_j(t)(v,v). \end{align*} Applying Gronwall's Inequality on $J$ gives \begin{align*} E_J^{-1}g_j(0) \le g_j(t_0) \le E_Jg_j(0). \end{align*} Here \begin{align*} E_J:=\exp(2nC_J|J|). \end{align*} This comparison holds on all of $M_j$. Passing to the smooth pointed limit on the exhausting domains $K_m$ yields the same comparison on all of $M_\infty$: \begin{align*} E_J^{-1}g_\infty(0) \le g_\infty(t_0) \le E_Jg_\infty(0). \end{align*} Thus every escaping piecewise $C^1$ curve has infinite $g_\infty(t_0)$-length, because its $g_\infty(t_0)$-length is at least $E_J^{-1/2}$ times its infinite $g_\infty(0)$-length. Hopf-Rinow therefore gives completeness of $g_\infty(t_0)$.[/guided]