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American Journal of Mathematics
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arXiv.org e-Print Archive
Preprint . 2004
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American Journal of Mathematics
Article . 2006 . Peer-reviewed
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https://dx.doi.org/10.48550/ar...
Article . 2004
License: arXiv Non-Exclusive Distribution
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https://dx.doi.org/10.1353/ajm...
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Article . 2006
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Differentiability of the stable norm in codimension one

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Feb 2006Embargo end date: 01 Jan 2004 Germany English Publisher:Project MUSEJournal:American Journal of Mathematics, volume 128, pages 215-238 (eissn: 1080-6377, Copyright policy )
Authors: Auer, Franz; Bangert, Victor;

doi: 10.1353/ajm.2006.0002 , 10.48550/arxiv.math/0403235

arXiv: math/0403235

Differentiability of the stable norm in codimension one

Abstract

The real homology of a compact, n -dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm on H n -1 ( M , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) is a homogenized version of the Riemannian ( n -1)-volume. We study the differentiability properties of the stable norm at points α ε H n -1 ( M , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]). They depend on the position of α with respect to the integer lattice H n -1 ( M , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]) in H n -1 ( M , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]). In particular, we show that the stable norm is differentiable at α if α is totally irrational.

Country
Germany
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Keywords

Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), FOS: Mathematics, 49Q20 (Primary) 35B27, 53C38 (Secondary), Analysis of PDEs (math.AP)

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
10
Average
Top 10%
Average
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bronze