Representing homotopy groups and spaces of maps by p-harmonic maps

We prove existence theorems representing homotopy classes by p-harmonic maps of least p-energy, and representing n(th)homotopy classes by n-harmonic maps, generalizing theorems of Cartan and Sacks-Uhlenbeck. This leads to a new generalized principle of Synge which gives a new role in Riemannian geometry to p-harmonic maps. Under the assumption of vanishing pi(n)(N), we prove the existence of n-harmonic maps in which the case n = 2 is due to Sacks-Uhlenbeck, Schoen-Yau, and Lemaire, and the case n greater than or equal to 2 also due to Jest by a different approach. We classify compact, irreducible, p-superstrongly unstable symmetric spaces (extending the work of Howard-Wei and Ohnita) and furnish an example of "gap phenomena" that do not arise for ordinary harmonic maps. Augmenting our previous existence theorem for p-harmonic maps on manifolds without boundary in spaces of maps, we solve the Dirichlet problem for p-harmonic maps in general dimensions that was solved by Hamilton in the case where p = 2 and the target manifold has non-positive sectional curvature. We also prove a uniqueness theorem for p-harmonic maps that generalizes the work of Hartman for p = 2.