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p-harmonic coordinates for Hölder metrics and applications

Abstract
We show that on any Riemannian manifold with H¨older continuous metric tensor, there exists a p-harmonic coordinate system near any point. When p = n this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having C α metric tensors is C 1+α regular, and that a manifold with W1,n ∩ C α metric tensor and with vanishing Weyl tensor is locally conformally flat if n ≥ 4. The results extend the works [LS14, LS15] from the case of C 1+α metrics to the H¨older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.
Main Authors
Julin, Vesa Liimatainen, Tony Salo, Mikko
Format
Articles Research article
Published
2017
Series
Communications in Analysis and Geometry
Subjects
p-harmonic coordinates
Hölder metrics
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/27147515
Publisher
International Press
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-201801121163Use this for linking
Review status
Peer reviewed
ISSN
1019-8385
DOI
https://doi.org/10.4310/CAG.2017.v25.n2.a5
Language
English
Published in
Communications in Analysis and Geometry
Citation
License
Open Access
Copyright© the Authors, 2017. This is a final draft version of an article whose final and definitive form has been published by International Press. Published in this repository with the kind permission of the publisher.

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author Julin, Vesa Liimatainen, Tony Salo, Mikko
author_facet Julin, Vesa Liimatainen, Tony Salo, Mikko
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description We show that on any Riemannian manifold with H¨older continuous metric tensor, there exists a p-harmonic coordinate system near any point. When p = n this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having C α metric tensors is C 1+α regular, and that a manifold with W1,n ∩ C α metric tensor and with vanishing Weyl tensor is locally conformally flat if n ≥ 4. The results extend the works [LS14, LS15] from the case of C 1+α metrics to the H¨older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.
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discipline_txtF Inversio-ongelmien huippuyksikkö
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spellingShingle p-harmonic coordinates for Hölder metrics and applications Julin, Vesa Liimatainen, Tony Salo, Mikko p-harmonic coordinates Hölder metrics Communications in Analysis and Geometry
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title p-harmonic coordinates for Hölder metrics and applications
title_full p-harmonic coordinates for Hölder metrics and applications
title_fullStr p-harmonic coordinates for Hölder metrics and applications
title_full_unstemmed p-harmonic coordinates for Hölder metrics and applications
title_short p-harmonic coordinates for Hölder metrics and applications
title_sort pharmonic coordinates for Hölder metrics and applications
topic p-harmonic coordinates Hölder metrics

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