Let $(M^n,g(t))$ be a Ricci flow on a closed manifold for $t\in[0,T]$. Let $\tau:[0,T]\to(0,\infty)$ be a smooth function satisfying $\partial_t\tau=-1$. Let $f:M\times[0,T]\to\mathbb R$ be a smooth function, and define $u:M\times[0,T]\to(0,\infty)$ by

\begin{align*} u=(4\pi\tau)^{-n/2}e^{-f} \end{align*}

Assume that this function $u$ solves the conjugate [heat equation](/page/Heat%20Equation) and has total mass $1$ for every $t$. Let $\mathcal W$ denote Perelman's W-functional, viewed as a map from triples $(h,\varphi,\sigma)$ consisting of a Riemannian metric $h$ on $M$, a smooth function $\varphi:M\to\mathbb{R}$, and a positive parameter $\sigma\in(0,\infty)$ to $\mathbb{R}$. Then

\begin{align*} \frac{d}{dt}\mathcal W[g(t),f(t),\tau(t)] =2\tau\int_M \left|\operatorname{Ric}_g+\operatorname{Hess}_{g} f-\frac{1}{2\tau}g\right|_g^2 (4\pi\tau)^{-n/2}e^{-f}\,d\mu_g \ge 0. \end{align*}