In this paper, the author discusses the eigenvalues and entropies under the
harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map
flow. We give an alternative proof of results for compact steady and expanding
harmonic-Ricci breathers. In the second part, we derive some monotonicity
formulas for eigenvalues of Laplacian under the harmonic-Ricci flow. Finally,
we obtain the first variation of the shrinker and expanding entropies of the
harmonic-Ricci flow.Comment: 37 pages; final version to appear in Pacific J. Mat
