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Regularity theory for $ 2$-dimensional almost minimal currents I: Lipschitz approximation


Authors: Camillo De Lellis, Emanuele Spadaro and Luca Spolaor
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 49N60, 49Q05, 49Q15
DOI: https://doi.org/10.1090/tran/6995
Published electronically: July 19, 2017
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Abstract: We construct Lipschitz $ Q$-valued functions which carefully approximate integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $ 2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $ 3$-dimensional area minimizing cones.


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Additional Information

Camillo De Lellis
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Emanuele Spadaro
Affiliation: Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
Email: Emanuele.Spadaro@math.uni-leipzig.de

Luca Spolaor
Affiliation: Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: lspolaor@mit.edu

DOI: https://doi.org/10.1090/tran/6995
Received by editor(s): October 22, 2015
Received by editor(s) in revised form: April 18, 2016, and June 7, 2016
Published electronically: July 19, 2017
Additional Notes: The first and third authors’ research was supported by the ERC grant RAM (Regularity for Area Minimizing currents), ERC 306247.
Article copyright: © Copyright 2017 American Mathematical Society