This is an outline for the Rutgers Grad Analysis seminar talk that I delivered on 4/25/2025.
Abstract: In this talk, we will define energy minimizing maps and introduce key examples. Then, we will derive the associated variational formulae. A central theme of the talk will be the monotonicity formula: we will explore its significance and highlight its appearance in related geometric contexts such as minimal surfaces, mean curvature flow, and Ricci flow. We will then prove the monotonicity formula for energy minimizing maps. Finally, we will define the density function, introduce tangent maps, and discuss the role of monotone quantities in the study of regularity and singularities.
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