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The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solutions and the Ricci flow. These results will be obtained using the methods of Geometric analysis and, in particular, due to theorems of Yau, Li and Schoen on the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions., Comment: arXiv admin note: substantial text overlap with arXiv:1801.06433