Monotonicity formulae and holomorphicity of harmonic maps between Kahler manifolds

In this paper, we introduce the stress-energy tensors of the partial energies E-'(f) and E-' '(f) of maps between Kahler manifolds. Assuming the domain manifolds possess some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps into any Kahler manifolds and harmonic maps into Kahler manifolds with strongly semi-negative curvature. These monotonicity inequalities enable us to derive some holomorphicity and Liouville-type results for these pluriharmonic maps and harmonic maps. We also use the stress-energy tensors to investigate the holomorphic extension problem of Cauchy-Riemann maps.