The Related Problems Of The Geometry Of Harmonic Map

Harmonic mapping is a natural extension of geodesic,the minimal submanifolds and harmonic function concepts in differential geometry.It is related to the analytical mapping in function theory of several complex variables,the random processes theory,the liquid crystal theory of materials science,the nonlinear field theory in the theoretical physical and so there are many geometers who are interested in harmonic maps.We study some relevant geometric properties of generalized harmonic maps throu-gh geometric analysis methods.Firstly,we derive the subgradient estimate for positive solutions to a nonlinear subelliptic equation in a complete pseudohermitian(2n + 1)-manifold by using the methods of CR sub-Laplacian comparison property and CR Bochner formula.Secondly,we introduce the notion of(weakly)quasi-F-harmonic map with potential from a smooth metric measure space to a Riemannian manifold.By using the stress-energy tensor,we prove some Liouville type results for these maps under conditions on H and the Bakry-′Emery Ricci tensor.Thirdly,we introduce a complete noncompact submanifold in a Hadamard manifold with the nonpositive sectional curvature.We obtain that some Liouville type results for p-harmonic functions.Finally,we use the stress-energy tensor to obtain some monotonicity formulas.By assuming some conditions on the asymptotic behavior of the maps at infinity,we obtain some Liouville type results for f-harmonic maps,F-Ginzburg-Landau energy and F-symphonic maps.