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Transactions of the American Mathematical Society

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

 
 

 

Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps


Authors: Jonas Hirsch, Salvatore Stuvard and Daniele Valtorta
Journal: Trans. Amer. Math. Soc. 371 (2019), 4303-4352
MSC (2010): Primary 58E20; Secondary 49Q20
DOI: https://doi.org/10.1090/tran/7595
Published electronically: November 2, 2018
MathSciNet review: 3917224
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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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Jonas Hirsch
Affiliation: Scuola Internazionale Superiore di Studi Avanzati, via Bonomea, 265, 34136 Trieste, Italy
Email: jonas.hirsch@sissa.it

Salvatore Stuvard
Affiliation: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: salvatore.stuvard@math.uzh.ch

Daniele Valtorta
Affiliation: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: daniele.valtorta@math.uzh.ch

DOI: https://doi.org/10.1090/tran/7595
Keywords: Q-valued functions, harmonic maps, singular set, rectifiability, Reifenberg theorem, quantitative stratification
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.
Article copyright: © Copyright 2018 American Mathematical Society