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arXiv · 2506.23608

Constraint Maps: Insights and Related Themes

Alessio Figalli·André Guerra·Sunghan Kim·Henrik Shahgholian
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Abstract

This paper explores recent progress related to constraint maps. Building on the exposition in [14], our goal is to provide a clear and accessible account of some of the more intricate arguments behind the main results in this work. Along the way, we include several new results of independent value. In particular, we give optimal geometric conditions on the target manifold that guarantee a unique continuation result for the projected image map. We also prove that the gradient of a minimizing harmonic map (or, more generally, of a minimizing constraint map) is an $A_\infty$-weight, and therefore satisfies a strong form of the unique continuation principle. In addition, we outline possible directions for future research and highlight several open problems that may interest researchers working on free boundary problems and harmonic maps.

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Alessio Figalli, André Guerra, Sunghan Kim, Henrik Shahgholian. 2025-06-30. Constraint Maps: Insights and Related Themes. https://arxiv.org/abs/2506.23608

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