  Let $\Omega$ be a compact Riemannian manifold with smooth boundary and let
$u_t$ be the solution of the heat equation on $\Omega$, having constant unit
initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the
boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is
a constant function on the boundary, we say that $\Omega$ has the {\it constant
flow property}. This gives rise to an overdetermined parabolic problem, and our
aim is to classify the manifolds having this property. In fact, if the metric
is analytic, we prove that $\Omega$ has the constant flow property if and only
if it is an {\it isoparametric tube}, that is, it is a solid tube of constant
radius around a closed, smooth, minimal submanifold, with the additional
property that all equidistants to the boundary (parallel hypersurfaces) are
smooth and have constant mean curvature. Hence, the constant flow property can
be viewed as an analytic counterpart to the isoparametric property. Finally, we
relate the constant flow property with other overdetermined problems, in
particular, the well-known Serrin problem on the mean-exit time function, and
discuss a counterexample involving minimal free boundary immersions into
Euclidean balls.
