Some Problems About Generalized Harmonic Maps

In this thesis, we investigate weakly F-harmonic maps with potential H, BMO conjec-ture, S. Meata’s Conjecture and generalized Chen’s conjecture.First, we introduce the notion of weakly F-harmonic maps with potential H with respect to the functional EF,H(u).By using the stress-energy tensor, we obtain some monotonicity formulas for these maps under conditions on H and the curvature conditions of the domain manifold. Then, some Liouville typ results are obtained by using these monotonicity formulas.We also investigate the complete biharmonic submanifolds in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant by using integral by parts. A complete biharmonic submanifolds in a Riemannian manifold has constant mean curvature if the integral of the mean curvature satisfies some growth conditions. So an affirmative partial answer to BMO conjecture and S. Meata’s Conjecture is obtained.In chapter 4 and 5 we investigate generalized Chen’s conjecture for p-biharmonic maps and F-biharmonic maps. We consider the nonexistence of p-biharmonic maps in nonpositively curved Riemannian manifolds and get Theorem 4.1 and Theorem 4.1. Finally, we consider the generalized Chen’s conjecture for F-biharmonic submanifold:any F-biharmonic submanifold in a Riemannian manifold with non-positive sectional curvature is minimal. By using integral estimate, we obtain that if F’>0,F"≥0 and T(u) satisfies some integral conditions, then it is minimal.we also obtain Theorem 4.1 when M is weakly convex F-biharmonic hypersurface in space form N(c). These results give affirmative partial answers to generalized Chen’s conjecture.