Rigidity Of V-harmonic Maps,Gradient Estimates Of A Parabolic Equation Concerning ?_V

This thesis spread around V-harmonic maps.We study Schwarz lemma,Liouville type theorem and constant boundary-problem.We also consider gradient estimates of parabolic equations concerning ?V.Chapter 1 is introduction,which simply introduces the research background.It also mainly summarize the structure of this thesis.In Chapter 2,we simply introduce the preliminaries of this thesis.It contains the connection,V-harmonic maps and holomorphic maps between almost Hermitian manifolds.Chapter 3 gives a Schwarz lemma of V-harmonic maps between Riemannian mani-folds.As applications,we obtain Schwarz lemma of the Weyl harmonic maps from con-formal Weyl manifolds to Riemannian manifolds and the holomorphic maps from almost Hermitain manifolds to quasi-Kahler manifolds.We also prove a volume-decreasing re-sult.In Chapter 4,we obtain that monotonicity formula and holomorphicity(anti-holomorphicity)of a class of maps between Hermitian manifolds by using the stress-energy tensors with respect to the partial energies.In Chapter 5,we show that a Liouville type theorem for V-harmonic maps between Riemannian manifolds by using the stress-energy tensor with respect to the energy.We also consider the constant boundary-value problem.In Chapter 6,we study Li-Yau type estimates and Souplet-Zhang type estimates of a parabolic equation concerning the operator ?V under the assumption with respect to the k-Bakry-Emery Ricci curvature RiOVk.Finally,in Chapter 7,we give some expectations for the further research.