$(M,g_t)$ is a family of Riemannian manifold ,$g_t$ evolve under Ricci flow $\partial_t g_{ij}=-2R_{ij}$. At $t=0$ ,we define $\varphi$ as below first picture .
Then ,extend $\varphi=0, $ outside $B_0(p,r)$ ,I am fuzzy with it.It means only $\varphi \ne 0$ when $(x,t)\in B_0(p,r)\times \{t:t=0\}$ or $\varphi\ne0 $ when $(x,t)\in B_0(p,r)\times (0,T]$. If it's the second situation, the $r $ is not independent of $t$, so ,the $\varphi$ is not independent of $t$. But if it's the first situation ,there is question that $\varphi$ is not continue about $t$, and the $H$ does not make sense.
I don't know how to understand it .
The below picture is from 194th page of this paper
