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On the Singular Set of Harmonic Maps into DM-Complexes


About this Title

Georgios Daskalopoulos and Chikako Mese

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 239, Number 1129
ISBNs: 978-1-4704-1460-3 (print); 978-1-4704-2741-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1129
Published electronically: June 4, 2015

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Harmonic maps into NPC spaces and DM-complexes
  • Chapter 3. Regular and Singular points
  • Chapter 4. Metric estimates near a singular point
  • Chapter 5. Assumptions
  • Chapter 6. The Target Variation
  • Chapter 7. Lower Order Bound
  • Chapter 8. The Domain variation
  • Chapter 9. Order Function
  • Chapter 10. The Gap Theorem
  • Chapter 11. Proof of Theorems \ref MAINTHEOREM–\ref GAPTHEOREM*
  • Appendix A. Appendix 1
  • Appendix B. Appendix 2

Abstract


We prove that the singular set of a harmonic map from a smooth Riemannian domain to a Riemannian DM-complex is of Hausdorff codimension at least two. We also explore monotonicity formulas and an order gap theorem for approximately harmonic maps. These regularity results have applications to rigidity problems examined in subsequent articles.

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