Liouville 's Theorem For Harmonic Mappings And Exponential Mappings

This chapter has two parts, investigating Liouville type properties for har-monic maps and exponentially harmonic maps respectively. In the first part, this chapter considers Liouville type prdperties in asymptotic conditions for harmonic maps as u:(M,g)→(N,h). Here, M is a complete and noncompact manifold, whose curvature satisfies certain pinching conditions. This enables us to get the monotonicity formula for harmonic maps implying a lower bound for the growth rate of the energy. On the other hand, the assumptions on the asymptotic limiting order at infinity imply an upper bound for the growth rate of the enegy. The two bounds are contradictionary in proper conditions unless the harmonic map is a constant map. In the second part, this chapter discusses exponentially harmonic maps. Firstly, we present the conservation law and the unique continuation theorem for exponentially harmonic maps. Lastly, we study the nonexistence of some particular exponentially harmonic maps in this form u:(Rm,f2g0)→(N,h) in asymptotic conditions, where go is the Euclidean metric on Rm, m≥3, and f is a smooth function on Rm.