The first eigenvalue of the p-Laplacian on time dependent Riemannian metrics

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Let (M,g) be an n-dimensional compact Riemannian manifold (n>1) whose metric g(t) evolves by the generalized abstract geometric flow. This paper discusses the evolution, monotonicity and differentiability for the first eigenvalue of the p-Laplacian on (M,g(t)) with respect to time evolution. We prove that the first nonzero p-eigenvalue is monotone nondecreasing along the flow under certain geometric condition and that the first eigenvalue is differentiable almost everywhere. When p=2, we recover the corresponding results for the usual Laplace-Beltrami operator. Our results provide a unified approach to the study of p-eigenvalue under various geometric flows