Some Results On Generalized Harmonic Maps And Harmonic Forms

We study several generalized harmonic maps and its properties through geometric analysis methods.The main contents include some vanishing and finiteness theorems for Lp p-harmonic 1-forms on a locally conformally flat Riemannian manifolds,some monotonicity formulas and Liouville type results for CC-stationary maps and subgradient estimates for a nonlinear subparabolic equation via nonnegative Bakry-Emery pseudohermitian Ricci curvature on complete noncompact pseudohermitian manifold.The thesis is divided into four chapters.In Chapter 1,we briefly introduce the history of the subject we studied.After that,our main results in this thesis are presented.In Chapter 2,we obtain some vanishing and finiteness theorems for Lp p-harmonic 1-forms on a locally conformally flat Riemmannian manifolds which satisfies an integral pinching condition on the traceless Ricci tensor,and for which the scalar curvature satisfies pinching curvature conditions or the first eigenvalue of the Laplace-Beltrami operator of M is bounded by a suitable constant.In Chapter 3,a horizontal functional ?H for maps from Riemannian manifolds to pseudo-Hermitian manifold is introduced.The critical maps of this functional are called CC-stationary maps.Then by using horizontal stress-energy tensor,some monotonicity formulas and Liouville type theorems are obtained for the horizontal CC-stationary maps from Riemannian manifolds to pseudo-Hermitian manifold or the CC-stationary maps from Riemannian manifold to sasakian manifold.In Chapter 4,firstly,some basic materials in a pseudohermitian(2n + 1)-manifold are intro-duced,then we consider a complete noncompact pseudohermitian(2n + 1)-manifold,and obtain the sub-gradient estimate for positive solution of a nonliner subparabolic equation on B(R)which satisfies the CR sub-Laplacian comparison property and the m-Bakry-Emery(or ?-Bakry-Emery)pseudoher-mitian Ricci curvature condition.