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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps
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by Jonas Hirsch, Salvatore Stuvard and Daniele Valtorta PDF
Trans. Amer. Math. Soc. 371 (2019), 4303-4352 Request permission

Abstract:

In this paper we study the singular set of energy minimizing $Q$-valued maps from $\mathbb {R}^m$ into a smooth compact manifold $\mathcal {N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single-valued case, we prove that the target $\mathcal {N}$ being nonpositively curved but not simply connected does not imply continuity of the map.
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Additional Information
  • Jonas Hirsch
  • Affiliation: Scuola Internazionale Superiore di Studi Avanzati, via Bonomea, 265, 34136 Trieste, Italy
  • MR Author ID: 1179776
  • Email: jonas.hirsch@sissa.it
  • Salvatore Stuvard
  • Affiliation: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  • MR Author ID: 1231535
  • ORCID: 0000-0002-3519-3653
  • Email: salvatore.stuvard@math.uzh.ch
  • Daniele Valtorta
  • Affiliation: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  • MR Author ID: 956785
  • Email: daniele.valtorta@math.uzh.ch
  • Received by editor(s): August 31, 2017
  • Received by editor(s) in revised form: March 20, 2018
  • Published electronically: November 2, 2018
  • Additional Notes: The research of the first author has been supported by the MIUR SIR-grant “Geometric Variational Problems”, ID RBSI14RVEZ
    The research of the second author was supported by the ERC-grant RAM “Regularity of Area Minimizing Currents”, ID 306246
    The research of the third author has been supported by the SNSF grant PZ00P2_168006.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 4303-4352
  • MSC (2010): Primary 58E20; Secondary 49Q20
  • DOI: https://doi.org/10.1090/tran/7595
  • MathSciNet review: 3917224