The Related Problems And The Geometry Of Generalized ?-Harmonic Map

Harmonic mapping is a natural extension of the concepts of geodesic,minimal submanifold and harmonic function in differential geometry.It is closely related to the holomorphic mapping in multicomplex function theory,random process theory and nonlinear field theory in theoretical physics,so it has been widely concerned by geometrists.We mainly study the geometry of generalized?-harmonic maps and some relevant properties by geometric analysis methods.The thesis includes some results for SET-p-stationary maps and CR gradient estimate for the positive eigenfunction of Witten sub-Laplacian.The thesis is divided into three chapters:In chapter 1,we introduce the backgrand,research significance,status of gener-alized?-harmonic maps and the main results we have obtained.In chapter 2,we introduce a functional?_p,Sfor the SET-p-stationary maps be-tween Riemannian manifolds.First,we deduce the variation formulas of the functional?_p,S.Then,by using the stress-energy tensor,we obtain some Liouville type theorems for SET-p-stationary maps.Finally,we prove that the stable SET-p-stationary maps from or into the compact convex hypersurface must be constant if the principal cur-vatures of compact convex hypersurface satisfy the inequality (2p-1)?_m<?_m-1i=1?_i.In chapter 3,under m-Bakry-(?)mery(or?-Bakry-(?)mery)pseudohermitian Ricci curvature condition,we derive subgradient estimate for positive solutions of Witten sub-Laplacian(?_H,?u(x)=-?u(x))on a weighted complete noncompact pseudoher-mitian manifold which satisfies the sub-Laplacian comparison property.