Fisrt and second pictures below is from Entropy and heat kernel bounds on a Ricci flow background, I don't know the mean of $f_{t_0-\tau}$. Acoording to second picture, I guess that $f_{t_0-\tau}$ is the $f(y,s)$ of $$ K(x_0,t_0-\tau;y,s) =(4\pi(t_0-\tau-s))^{-n/2}e^{-f(y,s)} \tag{1} $$ where $K(x,t;y,s)$ is a conjugate heat kernel based at $(x,t)$. But I am not sure, since I think the pointed at $(x_0,t_0)$ should mean that $f_{t_0-\tau}$ is the $f(y,s)$ of $$ K(x_0,t_0;y,s)=(4\pi(t_0-s))^{-n/2}e^{-f(y,s)} \tag{2} $$
Besides, from the New logarithmic Sobolev inequalities and an $\epsilon$-regularity theorem for the Ricci flow [NH14], the 3th picture below, I also think the $(1)$ is right.
So, I want to know what is the $f_{t_0-\tau}$ ?
