Subharmonicity of the Dirichlet energy and harmonic mappings from Kähler manifolds

In this thesis, we provide an application of a Bochner type formula of Siu and Sampson; our main result is as follows [1]: If { M_t } t∈∆ is a polarized family of compact Kähler manifolds over the open unit disk ∆, if N is a Riemannian manifold satisfying the curvature condition: R^N (X, Y, X, Y) ≤ 0 for X, Y ∈ T_C N, and if { φ_t : M_t → N } t∈∆ is a smooth family of pluriharmonic maps, then the Dirichlet energy E( φ_t ) is a subharmonic function of t ∈ ∆. We also investigate the two natural questions: Under what conditions is the energy E( φ_t ) strictly subharmonic? What type of families { φ_t } t∈∆ have constant energy? Some of our answers generalize the results of Tromba [2] and Toledo [3], which concern the case where M_t are compact Riemann surfaces. We conclude this thesis with a discussion of examples of subharmonicity of the energy.