REPRESENTING HOMOTOPY-GROUPS AND SPACES OF MAPS BY P-HARMONIC MAPS

We prove existence theorems representing homotopy classes by p-harmoni c maps of least p-energy, and representing n(th)homotopy classes by n- harmonic maps, generalizing theorems of Cartan and Sacks-Uhlenbeck. Th is leads to a new generalized principle of Synge which gives a new rol e in Riemannian geometry to p-harmonic maps. Under the assumption of v anishing 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 fur nish an example of ''gap phenomena'' that do not arise for ordinary ha rmonic maps. Augmenting our previous existence theorem for p-harmonic maps on manifolds without boundary in spaces of maps, we solve the Dir ichlet problem for p-harmonic maps in general dimensions that was solv ed 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.