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We study the evolution and monotonicity of the eigenvalues of  p -Laplace operator on an  m -dimensional compact Riemannian manifold  M  whose metric  g ( t ) evolves by the Ricci-harmonic flow. The first nonzero eigenvalue is proved to be monotonically nondecreasing along the flow and differentiable almost everywhere. As a corollary, we recover the corresponding results for the usual Laplace–Beltrami operator when  p  = 2. We also examine the evolution and monotonicity under volume preserving flow and it turns out that the first eigenvalue is not monotone in general.

Evolution and monotonicity of the first eigenvalue of p -Laplacian under the Ricci-harmonic flow