Some Study On Polynomial ∞-Harmonic And Biharmonic Maps

In this thesis, we study linear∞-harmonic and biharmonc maps between Riema-nian manifolds. We give complete classifications of linear∞-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all∞-harmonic linear endomorphisms of Sol space and show that there is a subgroup of∞-harmonic linear automorphisms in the group of linear automorphisms of Sol space.While working on the thesis the author was lucky to have the opportunity to work with Professor Ye-Lin Ou on classifications of∞-harmonic and biharmonic maps between Riemannian manifolds. The results obtained have appeared in [WO] and [OW1] in which we give complete classifications of linear and quadratic∞-harmonic maps from and into a sphere, quadratic∞-harmonic maps between Euclidean spaces. We describe all linear and quadratic∞-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic∞-harmonic maps between the Euclidean spaces and complex Euclidean spaces. We also study biharmonic maps into Sol, Nil and Heisenberg spaces. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Euclidean space into Sol, Nil, or Heisenberg space is biharmonic if and only if it is a harmonic map, and give a complete classification of such maps.