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

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