We study the evolution and monotonicity of the eigenvalues of

This paper studies L p-version of the Hardy type inequalities on the geodesic sphere of constant sectional curvature and establishes that the corresponding constant is sharp. Furthermore, the inequalities obtained are used to derive an uncertainty principle inequality and another inequality involving the first nonzero eigenvalue of the p-Laplacian on the sphere.

Let ∆ ϕ = ∆ − ∇ϕ∇ be a symmetric diffusion operator with an invariant weighted volume measure dµ = e −ϕ dv on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of ∆ ϕ and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case n ≥ 3. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.MSC (2010): Primary 53C21, 53C44, Secondary 35P30, 58J35

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