<p dir="ltr">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<sub>t</sub> } 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<sup>N</sup> (X, Y, X, Y) ≤ 0 for X, Y ∈ T<sub>C</sub> N, and if { φ<sub>t </sub>: M<sub>t</sub> → N } t∈∆ is a smooth family of pluriharmonic maps, then the Dirichlet energy E( φ<sub>t</sub> ) is a subharmonic function of t ∈ ∆. We also investigate the two natural questions: Under what conditions is the energy E( φ<sub>t</sub> ) strictly subharmonic? What type of families { φ<sub>t</sub> } t∈∆ have constant energy? Some of our answers generalize the results of Tromba [2] and Toledo [3], which concern the case where M<sub>t</sub> are compact Riemann surfaces. We conclude this thesis with a discussion of examples of subharmonicity of the energy.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/27679503 |
| Date | 12 November 2024 |
| Creators | Che-Hung Huang (20175057) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | In Copyright |
| Relation | https://figshare.com/articles/thesis/Subharmonicity_of_the_Dirichlet_energy_and_harmonic_mappings_from_K_hler_manifolds/27679503 |
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