  Energy minimizing maps (E.M.M.s) play a central role in the calculus of
variations, partial differential equations (PDEs), and geometric analysis.
These maps are often embedded into $C^\infty$ Riemannian manifolds to minimize
the Dirichlet Energy functional under certain prescribed conditions. For
understanding physical phenomena where systems naturally evolve to states of
minimal energy, the geometric analysis of these maps has provided elucidating
insights. This paper explores the geometric and analytic properties of energy
minimizing maps, tangent maps, and the singular set (sing(u)). We begin by
establishing key concepts from analysis, including the Sobolev Space $W^{1,2}$
harmonic functions, and Hausdorff dimension. Significant results about the
density function, its upper semi-continuity, and the compactness theorem for
tangent maps, and theorems for homogeneous degree zero minimizers are
presented. Also analyzed in detail is the singular set (sing(u)), its Hausdorff
dimension, and geometric structure. We conclude with open problems that are
rich in research potential, and the far-reaching implications if these problems
are to be solved.
