[proofplan] We write the reduced distance using the relation \begin{align*} l=\frac{L}{2\sqrt{\tau}}. \end{align*} Here $L$ is the minimizing $\mathcal L$-length from $(p,0)$ to $(q,\tau)$. At a smooth point reached by a minimizing $\mathcal L$-geodesic, Perelman's first variation identities give formulas for $\partial_\tau l$ and $|\nabla l|^2$, while the second variation along a trace family of space-time Jacobi fields gives the required upper bound for $\Delta l$. Combining these three identities cancels the auxiliary curvature integral exactly and yields the differential inequality. Across the $\mathcal L$-cut locus, the same calculation is applied to smooth upper support functions obtained by freezing a minimizing segment before the endpoint and appending a short terminal geodesic; letting the frozen time approach the endpoint gives the weak upper-barrier inequality. [/proofplan] [step:Declare the reduced length notation and the minimizing curve] Let $L: M \times (0,t_0] \to \mathbb{R} \cup \{\infty\}$ denote the reduced length based at $(p,0)$, so that \begin{align*} l(x,\sigma)=\frac{L(x,\sigma)}{2\sqrt{\sigma}} \end{align*} for every $(x,\sigma) \in M \times (0,t_0]$ with $L(x,\sigma)<\infty$. Fix a point $(q,\tau)$ satisfying the hypotheses. Let \begin{align*} \gamma:[0,\tau] &\to M \end{align*} be a minimizing $\mathcal L$-geodesic from $p$ to $q$, with terminal velocity $X \in T_qM$ defined by $X:=\dot{\gamma}(\tau)$. For each $\varepsilon\in(0,\tau)$, write $\gamma_\varepsilon:=\gamma|_{[\varepsilon,\tau]}$. The curvature bound on compact subintervals makes the first and second variation formulae for $\mathcal L$ valid along $\gamma_\varepsilon$. We use the [reduced length endpoint variation and initial asymptotic estimates](/page/Reduced%20Distance) for complete backward Ricci flows with bounded curvature on compact positive-time subintervals. Applied to the fixed minimizing curve $\gamma$, this result supplies constants $C_0>0$ and $s_0\in(0,\tau)$ such that, for $0<s<s_0$, $\sqrt{s}\,|\dot\gamma(s)|_{g(s)}\leq C_0$, and every weighted curvature expression appearing in the first variation and traced second variation is bounded in absolute value by $C_0(1+s^{-1/2})$. These dominating functions belong to $L^1((0,s_0),\mathcal L^1)$, while all terms are bounded on $[s_0,\tau]$ by the curvature hypothesis on compact positive-time intervals. Hence the identities obtained on $[\varepsilon,\tau]$ pass to the limit by dominated convergence as $\varepsilon\downarrow0$. [/step] [step:Record the first variation identities at the smooth endpoint] Define the scalar quantity $K \in \mathbb{R}$ along $\gamma$ by \begin{align*} K:=\int_0^\tau s^{3/2}\Bigl(\partial_s R(\gamma(s),s)+2\nabla R(\gamma(s),s)\cdot \dot{\gamma}(s)-2\operatorname{Ric}_{g(s)}(\dot{\gamma}(s),\dot{\gamma}(s))\Bigr)\,d\mathcal{L}^1(s), \end{align*} where $R(\cdot,s)$ is the scalar curvature of $g(s)$ and $\nabla R(\cdot,s)$ is its gradient with respect to $g(s)$. Since $l$ is smooth at $(q,\tau)$ and $\gamma$ is minimizing, the [Perelman first variation formula for reduced length](/page/Reduced%20Distance) applies. Its hypotheses are the backward Ricci flow equation, bounded curvature on compact positive-time intervals, a minimizing $\mathcal L$-geodesic ending at the differentiability point, and the limiting domination from $[\varepsilon,\tau]$ to $[0,\tau]$ established above. It gives the following three identities. First, \begin{align*} \nabla l(q,\tau)=X. \end{align*} Second, \begin{align*} |\nabla l|^2(q,\tau)+R(q,\tau)=\frac{l(q,\tau)}{\tau}-\frac{K}{\tau^{3/2}}. \end{align*} Third, \begin{align*} \partial_\tau l(q,\tau)=-\frac{l(q,\tau)}{\tau}+R(q,\tau)+\frac{K}{2\tau^{3/2}}. \end{align*} Here the gradient and norm are taken with respect to the metric $g(\tau)$. [guided] The first variation step translates the variational definition of reduced length into differential information about $l$. The quantity $K$ is introduced only to keep track of the curvature terms produced when the endpoint time varies; it will cancel after the Laplacian estimate is inserted. More explicitly, for the minimizing $\mathcal L$-geodesic $\gamma:[0,\tau]\to M$ ending at $q$, define \begin{align*} K:=\int_0^\tau s^{3/2}\Bigl(\partial_s R(\gamma(s),s)+2\nabla R(\gamma(s),s)\cdot \dot{\gamma}(s)-2\operatorname{Ric}_{g(s)}(\dot{\gamma}(s),\dot{\gamma}(s))\Bigr)\,d\mathcal{L}^1(s). \end{align*} The curvature boundedness on compact subintervals ensures that the variation formulae may be applied first on $[\varepsilon,\tau]$. The missing point is the limit as $\varepsilon\downarrow0$: the standard minimizing-curve asymptotic gives constants $C_0>0$ and $s_0\in(0,\tau)$ such that the velocity terms are bounded by $C_0s^{-1/2}$ and the weighted curvature terms are bounded by $C_0(1+s^{-1/2})$ on $(0,s_0)$. These bounds are integrable with respect to $\mathcal L^1$ on $(0,s_0)$, and the curvature hypothesis controls the remaining interval $[s_0,\tau]$. Thus dominated convergence passes the identities from $[\varepsilon,\tau]$ to $[0,\tau]$. Since $(q,\tau)$ is assumed to be a smooth point of $l$, the minimizing endpoint variation differentiates the infimum without ambiguity. Perelman's [first variation formula](/theorems/2728) therefore gives the spatial gradient identity \begin{align*} \nabla l(q,\tau)=\dot{\gamma}(\tau)=X. \end{align*} It also gives the Hamilton-Jacobi identity \begin{align*} |\nabla l|^2(q,\tau)+R(q,\tau)=\frac{l(q,\tau)}{\tau}-\frac{K}{\tau^{3/2}}. \end{align*} Finally, varying the terminal time gives \begin{align*} \partial_\tau l(q,\tau)=-\frac{l(q,\tau)}{\tau}+R(q,\tau)+\frac{K}{2\tau^{3/2}}. \end{align*} These identities are the reduced-distance analogues of differentiating an ordinary squared distance function along a minimizing geodesic. The extra term $K$ appears because the metric itself evolves by the backward Ricci flow. [/guided] [/step] [step:Estimate the Laplacian by the second variation trace] Let $\{e_1,\dots,e_n\}$ be a $g(\tau)$-[orthonormal basis](/page/Orthonormal%20Basis) of $T_qM$. For each $i\in\{1,\dots,n\}$ and each $\varepsilon\in(0,\tau)$, let $Y_{i,\varepsilon}:[\varepsilon,\tau]\to TM$ be the $\mathcal L$-Jacobi field along $\gamma_\varepsilon$ with $Y_{i,\varepsilon}(\varepsilon)=0$ and $Y_{i,\varepsilon}(\tau)=e_i$. The singular initial condition for the limiting field means precisely that $Y_i(s):=\lim_{\varepsilon\downarrow0}Y_{i,\varepsilon}(s)$ exists for $s\in(0,\tau]$, satisfies $Y_i(\tau)=e_i$, and obeys $|Y_i(s)|_{g(s)}=O(\sqrt{s})$ as $s\downarrow0$. Because $(q,\tau)$ is a smooth point of $L$, the second spatial derivative of $L(\cdot,\tau)$ at $q$ is computed by differentiating endpoint variations. The fields $Y_{i,\varepsilon}$ are the solutions of the $\mathcal L$-Jacobi equation, namely the linearized $\mathcal L$-geodesic equation along $\gamma_\varepsilon$, with the stated endpoint values. The [traced $\mathcal L$-Jacobi second variation estimate](/page/Reduced%20Distance) applies on $[\varepsilon,\tau]$ because the curvature is bounded on that compact positive-time interval and the endpoint fields are admissible. It gives an upper bound for the second derivative of $L(\cdot,\tau)$ in the direction $e_i$; summing over the $g(\tau)$-orthonormal basis gives the trace. The same dominated-convergence argument used for the first variation, together with the asymptotic $|Y_i(s)|_{g(s)}=O(\sqrt{s})$ supplied by the reduced length Jacobi-field asymptotic estimate, passes the traced second variation to the singular initial endpoint. The traced estimate yields \begin{align*} \Delta L(q,\tau)\leq \frac{n}{\sqrt{\tau}}-2\sqrt{\tau}R(q,\tau)-\frac{K}{\tau}. \end{align*} Since the spatial Laplacian is taken at fixed time $\tau$ and \begin{align*} l=\frac{L}{2\sqrt{\tau}}, \end{align*} this becomes \begin{align*} \Delta l(q,\tau)\leq \frac{n}{2\tau}-R(q,\tau)-\frac{K}{2\tau^{3/2}}. \end{align*} [guided] The second variation step is the place where the Laplacian enters. Choose a $g(\tau)$-orthonormal basis $\{e_1,\dots,e_n\}$ of $T_qM$. For each $i\in\{1,\dots,n\}$ and each $\varepsilon\in(0,\tau)$, let $Y_{i,\varepsilon}:[\varepsilon,\tau]\to TM$ be the $\mathcal L$-Jacobi field along $\gamma_\varepsilon$ with $Y_{i,\varepsilon}(\varepsilon)=0$ and $Y_{i,\varepsilon}(\tau)=e_i$. The limiting singular Jacobi field $Y_i:(0,\tau]\to TM$ is defined by \begin{align*} Y_i(s):=\lim_{\varepsilon\downarrow0}Y_{i,\varepsilon}(s), \end{align*} and satisfies $Y_i(\tau)=e_i$ and $|Y_i(s)|_{g(s)}=O(\sqrt{s})$ as $s\downarrow0$. Because $(q,\tau)$ is a smooth point of $L$, the Hessian of $L(\cdot,\tau)$ at $q$ is computed by differentiating endpoint variations through $q$. The [second variation formula](/theorems/2729) on $[\varepsilon,\tau]$ applies to the admissible fields $Y_{i,\varepsilon}$ because the curvature is bounded on that compact positive-time interval and the endpoints of each field are fixed as required. Summing the resulting estimates over the orthonormal basis takes the trace and therefore estimates $\Delta L(q,\tau)$. The passage from $[\varepsilon,\tau]$ to $[0,\tau]$ is justified by the same domination mechanism used in the first variation step, now with the Jacobi-field asymptotic included. Near $s=0$, the bounds $|\dot\gamma(s)|_{g(s)}=O(s^{-1/2})$ and $|Y_i(s)|_{g(s)}=O(\sqrt{s})$ make the traced second-variation terms bounded by an $L^1((0,s_0),\mathcal L^1)$ function of the form $C_1(1+s^{-1/2})$, for constants $C_1>0$ and $s_0\in(0,\tau)$. On $[s_0,\tau]$ all coefficients are bounded by the compact-time curvature hypothesis. Dominated convergence therefore gives the limiting traced $\mathcal L$-Jacobi estimate \begin{align*} \Delta L(q,\tau)\leq \frac{n}{\sqrt{\tau}}-2\sqrt{\tau}R(q,\tau)-\frac{K}{\tau}. \end{align*} Finally, at fixed terminal time $\tau$ the factor $(2\sqrt{\tau})^{-1}$ is spatially constant, so the relation \begin{align*} l=\frac{L}{2\sqrt{\tau}} \end{align*} gives \begin{align*} \Delta l(q,\tau)\leq \frac{n}{2\tau}-R(q,\tau)-\frac{K}{2\tau^{3/2}}. \end{align*} [/guided] [/step] [step:Combine the three formulas and cancel the curvature integral] Using the Laplacian estimate just obtained, we compute at $(q,\tau)$. First, \begin{align*} \partial_\tau l-\Delta l+|\nabla l|^2-R+\frac{n}{2\tau} \geq \partial_\tau l-\left(\frac{n}{2\tau}-R-\frac{K}{2\tau^{3/2}}\right)+|\nabla l|^2-R+\frac{n}{2\tau}. \end{align*} Therefore \begin{align*} \partial_\tau l-\Delta l+|\nabla l|^2-R+\frac{n}{2\tau}\geq \partial_\tau l+|\nabla l|^2+\frac{K}{2\tau^{3/2}}. \end{align*} Substituting the first variation identities gives \begin{align*} \partial_\tau l+|\nabla l|^2+\frac{K}{2\tau^{3/2}}=\left(-\frac{l}{\tau}+R+\frac{K}{2\tau^{3/2}}\right)+|\nabla l|^2+\frac{K}{2\tau^{3/2}}. \end{align*} We now use the identity \begin{align*} |\nabla l|^2+R=\frac{l}{\tau}-\frac{K}{\tau^{3/2}}. \end{align*} It follows that \begin{align*} \partial_\tau l+|\nabla l|^2+\frac{K}{2\tau^{3/2}}=\left(|\nabla l|^2+R\right)-\frac{l}{\tau}+\frac{K}{\tau^{3/2}}. \end{align*} Hence \begin{align*} \partial_\tau l+|\nabla l|^2+\frac{K}{2\tau^{3/2}}=\left(\frac{l}{\tau}-\frac{K}{\tau^{3/2}}\right)-\frac{l}{\tau}+\frac{K}{\tau^{3/2}}. \end{align*} Thus \begin{align*} \partial_\tau l+|\nabla l|^2+\frac{K}{2\tau^{3/2}}=0. \end{align*} This proves the asserted differential inequality at every smooth point reached by a minimizing $\mathcal L$-geodesic. [/step] [step:Pass the inequality to the cut locus in the weak upper-barrier sense] Let $(q,\tau)\in M\times(0,t_0]$ be a point with $l(q,\tau)<\infty$, possibly in the $\mathcal L$-cut locus. The weak upper-barrier formulation means the following: for every $\delta>0$ there is a smooth function $\phi$ defined in a neighbourhood of $(q,\tau)$ such that $\phi\geq l$ near $(q,\tau)$, $\phi(q,\tau)=l(q,\tau)$, and \begin{align*} \partial_\tau \phi(q,\tau)-\Delta \phi(q,\tau)+|\nabla \phi|^2(q,\tau)-R(q,\tau)+\frac{n}{2\tau}\geq -\delta. \end{align*} This is the [Calabi upper-barrier regularization argument for reduced distance](/page/Reduced%20Distance). One fixes a minimizing $\mathcal L$-geodesic to $(q,\tau)$, freezes the minimizing segment up to a small positive time $\varepsilon$, appends the smoothly varying terminal minimizing $\mathcal L$-geodesic on $[\varepsilon,\tau]$ for nearby endpoints, and then lets $\varepsilon\downarrow0$. Because $l$ is defined by an infimum of $\mathcal L$-lengths, the resulting fixed-path comparison length is greater than or equal to $l$ near $(q,\tau)$; after normalization by $2\sqrt{\tau}$ it gives smooth upper support functions $\phi_\varepsilon$ with $\phi_\varepsilon\geq l$ near $(q,\tau)$ and $\phi_\varepsilon(q,\tau)=l(q,\tau)$. The first-variation, second-variation, and traced $\mathcal L$-Jacobi quantities for $\phi_\varepsilon$ converge to the corresponding limiting quantities along the original minimizing curve by the same reduced length variation estimates already used above: the curvature terms are controlled on compact positive-time intervals, while the singular initial endpoint contributes integrable terms because $\sqrt{s}\,|\dot\gamma(s)|_{g(s)}=O(1)$ and the upper-barrier Jacobi fields satisfy $|Y_i(s)|_{g(s)}=O(\sqrt{s})$. Consequently there is an error function $\omega:(0,\tau)\to[0,\infty)$, defined by the absolute difference between the regularized variation trace and its limiting trace, such that $\lim_{\varepsilon\downarrow0}\omega(\varepsilon)=0$ and \begin{align*} \partial_\tau \phi_\varepsilon(q,\tau)-\Delta \phi_\varepsilon(q,\tau)+|\nabla \phi_\varepsilon|^2(q,\tau)-R(q,\tau)+\frac{n}{2\tau}\geq -\omega(\varepsilon). \end{align*} Given $\delta>0$, choose $\varepsilon\in(0,\tau)$ with $\omega(\varepsilon)<\delta$ and set $\phi:=\phi_\varepsilon$. This supplies the required smooth upper barrier and proves the weak upper-barrier inequality \begin{align*} \partial_\tau l-\Delta l+|\nabla l|^2-R+\frac{n}{2\tau}\geq0 \end{align*} wherever $l$ is finite. The proof is complete. [guided] At a cut-locus point $l$ need not be smooth, so the pointwise computation cannot be applied directly to $l$. The comparison construction available for reduced distance gives upper, not lower, supports: a fixed admissible path has $\mathcal L$-length at least the infimum defining $L$. Thus the valid weak formulation here is the weak upper-barrier formulation. For each $\delta>0$, we must find a smooth function $\phi$ near $(q,\tau)$ such that $\phi\geq l$, $\phi(q,\tau)=l(q,\tau)$, and \begin{align*} \partial_\tau \phi(q,\tau)-\Delta \phi(q,\tau)+|\nabla \phi|^2(q,\tau)-R(q,\tau)+\frac{n}{2\tau}\geq -\delta. \end{align*} Fix a minimizing $\mathcal L$-geodesic $\gamma:[0,\tau]\to M$ ending at $q$. For $\varepsilon\in(0,\tau)$, freeze the minimizing segment up to time $\varepsilon$ and define a local comparison function $\phi_\varepsilon$ by appending, from nearby endpoints, the smoothly varying terminal minimizing $\mathcal L$-geodesic on $[\varepsilon,\tau]$. The [Calabi upper-barrier regularization argument for reduced distance](/page/Reduced%20Distance) applies because the terminal interval lies in positive time, the curvature is bounded on compact positive-time subintervals, and the terminal endpoint variation is away from the singular initial time. Since $l$ is the infimum over all admissible $\mathcal L$-curves, this fixed-path comparison satisfies $\phi_\varepsilon\geq l$ near $(q,\tau)$. It also satisfies $\phi_\varepsilon(q,\tau)=l(q,\tau)$ because the frozen segment and the terminal segment together recover the chosen minimizing curve at the base endpoint. Now the smooth-point computation applies to $\phi_\varepsilon$ on $[\varepsilon,\tau]$. The first variation, second variation, and traced $\mathcal L$-Jacobi terms converge to the limiting quantities along the original minimizing curve as $\varepsilon\downarrow0$. The domination is the same as before: $\sqrt{s}\,|\dot\gamma(s)|_{g(s)}=O(1)$ controls the velocity singularity, the upper-barrier Jacobi fields satisfy $|Y_i(s)|_{g(s)}=O(\sqrt{s})$, and the weighted curvature and Jacobi terms are bounded by an $L^1((0,s_0),\mathcal L^1)$ function $C_2(1+s^{-1/2})$ near $0$, while compact positive-time curvature bounds control $[s_0,\tau]$. Define $\omega:(0,\tau)\to[0,\infty)$ to be the absolute difference between the regularized variation trace and its limiting trace. The convergence just described gives \begin{align*} \lim_{\varepsilon\downarrow0}\omega(\varepsilon)=0. \end{align*} For each $\varepsilon\in(0,\tau)$, the regularized computation gives \begin{align*} \partial_\tau \phi_\varepsilon(q,\tau)-\Delta \phi_\varepsilon(q,\tau)+|\nabla \phi_\varepsilon|^2(q,\tau)-R(q,\tau)+\frac{n}{2\tau}\geq -\omega(\varepsilon). \end{align*} Given $\delta>0$, choose $\varepsilon\in(0,\tau)$ such that $\omega(\varepsilon)<\delta$ and set $\phi:=\phi_\varepsilon$. This is exactly the required smooth upper barrier, so the inequality holds in the weak upper-barrier sense wherever $l$ is finite. [/guided] [/step]