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Monotonicity formulas for the first eigenvalue of the weighted p-Laplacianunder the Ricci-harmonic flow

Let p,φbe the weighted p-Laplacian defined on a smooth metric measure space.We study the evolution and monotonicity formulas for the first eigenvalue,λ1=λ(p,φ), of p,φunder the Ricci-harmonic flow. We derive some monotonicquantities involving the first eigenvalue, and as a consequence, this shows that λ1ismonotonically nondecreasing and almost everywhere differentiable along the flowexistence

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