For each $\tau \in (0,\tau_2]$, let $\nabla^{g(\tau)}$ denote the Levi-Civita connection of $g(\tau)$, let $R(\cdot,\tau):M \to \mathbb{R}$ denote the scalar curvature of $g(\tau)$, and let $\operatorname{Ric}_{g(\tau)}$ denote the Ricci tensor of $g(\tau)$.
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Let
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\begin{align*}
\gamma:(0,\tau_2] \to M
\end{align*}
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be a smooth $\mathcal L$-geodesic, meaning that $\gamma$ is a critical point, with fixed endpoints on every compact subinterval $[a,b] \subset (0,\tau_2]$, of Perelman's $\mathcal L$-length functional