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We estimate the heat kernel on a closed Riemannian manifold $M$, with$dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the resultdepends on some constants arising from a Sobolev imbedding theorem. In aspecial case, when the scalar curvature satisfies a certain natural inequality,we obtain, as a corollary, a bound similar to the one known for the fixedmetric case.

On the Heat Kernel under the Ricci Flow Coupled with the Harmonic Map Flow