  Given a solution of the (backwards) Ricci flow one can construct a so called
canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P.
Topping. We observe that for a mean curvature flow within a (backwards) Ricci
flow background, the space-time track of the mean curvature flow yields a
canonical soliton of the coupled flow within the canonical Ricci soliton. We
show that this provides a link between Hamilton's differential Harnack estimate
for the mean curvature flow and Hamilton's differential Harnack estimate for
the Ricci flow. Moreover the second fundamental form of our canonical soliton
matches the boundary term of the evolution of J. Lott's modified F-functional
for a Ricci flow with boundary. This functional also appears in quantum
gravity.
