The properties of P-harmonic maps and the application to Geometry

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Let u : M → N be a smooth map between Riemannian manifolds and p a real number 1 < p < ∞. We call u a p-harmonic map if it is a critical point of the p-energy functional ∫_M | du |^pdx. In the case of p = 2, it becomes the usual harmonic map. When N is a real number, the map u becomes the p-harmonic function and it is the solution of Δ_pu =div(|∇u|^<p-2>∇u) = 0. When M is the n-dimensional sphere S^n and p is equal to the dimension of M (dim M = n = p), we can get the existence of n-harmonic maps from S^n to N. This is the generalization of the results of Sacks-Uhlenbeck, which is the case of n = p = 2.Let N be a real number. For the p-Laplacian Δ_p, we define the first eigenvalue of the p-Laplacian as the least real number λ for which the equation Δ_pu = -λ|u|^<p-2>u has a nontrivial solution u. Before, we had several estimates for them on Riemannian manifolds, such as the Faber-Krahn type inequality, the Cheeger type inqulity, and the Cheng type inequality. We get a discrete analogue in this project term, that is, we define the p-Laplacian on graphs and get the Cheeger type inequality and the Brooks type inequality. Let G_1 = (V_1, E_1) and G_2 = (V_2, E_2) be two graphs and φ : V_1 → V_2 an onto mapping. The map φ is said to be a p-harmonic morphism of G_1 to G_2 if for any p-harmonic function f at y = φ(x) ∈ V_2, the composition φ^* f = f ο φ is p-harmonic function at x ∈ V_1. We show the p-harmonic morphism is equivalent to the horizontally conformal.Next we consider the solution of p-Laplace equations which coincide with Green kernels in the case of p = 2 and give some estimates.