Let Ω be an open set in a geodesically complete, non-compact, m-dimen-sional Riemannian manifold M with non-negative Ricci curvature, and without boundary. We study the heat flow from Ω into M−Ω if the initial temperature distribution is the characteristic function of Ω
Nearby in the stack
. We obtain a necessary and sufficient condition which ensures that an open set
Ω
with infinite measure has finite heat content for all
t>0
. We also obtain upper and lower bounds for the heat content of
Ω
in
M
. Two-sided bounds are obtained for the heat loss of