Picture below are from Entropy and heat kernel bounds on a Ricci flow background. I don't know how to use parabolic rescaling and time-shift to assume $r=1,t=1$.
$M$ is a manifold, and $g_t$ is Ricci flow.
$H_n$-center: A point $(z,t)\in M\times I$ is called an $H_n$-center of a point $(x_0,t_0)\in M\times I$ if $t<t_0$ and $$ Var_t(\delta_z, \nu_{x_0,t_0~;t})\le H_n(t_0-t) $$ where $H_n$ is a constant, and $$ Var_t(\delta_z, \nu_{x_0,t_0;t}) =\int_M\int_M d^2(x_1,x_2) \delta_z(x_1) dg_t(x_1) d\nu_{x_0,t_0~;t}(x_2) $$ is the variance of measures. $d\nu_{x_0,t_0~;t}$ is conjugate heat kernel measure, namely, denoting $d\nu_{x_0,t_0~;t}=K(x_0,t_0;\cdot, t) dg_t$, and $\Box^*=-\partial_t -\Delta_{g_t} +R$, then $$ \Box_{x,t}^* K(x_0,t_0; x, t) =0,~~~\lim_{t\rightarrow t_0^-}K(x_0,t_0; x, t) =\delta_{x_0}(x) $$
