On the Singular Sets of Harmonic Maps into F-Connected Complexes

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Date
2023-03-16
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Johns Hopkins University
Abstract
This work studies the singular sets of energy-minimizing maps u : Ω → X, where Ω is a Euclidean domain and X is an F-connected complex. This study builds on the work of Gromov and Schoen, in which a notion of energy-minimizing maps into singular spaces is developed, and F-connected complexes and the singular sets of maps into them are defined. In their work, it is shown that such singular sets are closed sets of Hausdorff dimension (m − 2), using analogues of some classical tools such as Almgren’s frequency function and its monotonicity formula. More recent work, especially a series of papers of Naber and Valtorta, and two papers of Azzam and Tolsa, provides a number of tools to use such monotone quantities to prove stronger results. The work herein adapts these more recent methods to show that the singular set of an energy-minimizing map u : Ω → X is (m − 2)-rectifiable, where Ω is an m-dimensional Euclidean domain and X is an F -connected complex.
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Keywords
Harmonic Maps, Regularity of Solutions, Rectifiability, Singular Spaces
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