  The paper establishes a series of gradient estimates for positive solutions
to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled
with the harmonic map flow between $M$ and a second manifold $N$. We prove
Li-Yau type Harnack inequalities and we consider the cases when $M$ is a
complete manifold without boundary and when $M$ is compact, without boundary.

Differential Geometry