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Curves Between Lipschitz and C 1 and Their Relation to Geometric Knot Theory
AuthorBlatt, Simon
Published in
The Journal of Geometric Analysis, New York, 2018, Vol. 2018, page 1-23
PublishedNew York : Springer US, 2018
Document typeJournal Article
Keywords (EN)Vanishing mean oscillation / Möbius energy / Gamma convergence
URNurn:nbn:at:at-ubs:3-10463 Persistent Identifier (URN)
 The work is publicly available
Curves Between Lipschitz and C 1 and Their Relation to Geometric Knot Theory [0.51 mb]
Abstract (English)

In this article, we investigate regular curves whose derivatives have vanishing mean oscillations. We show that smoothing these curves using a standard mollifier one gets

regular curves again. We apply this result to solve a couple of open problems. We show that curves with finite Möbius energy can be approximated by smooth curves in the energy space W 3 2,2 such that the energy converges which answers a question of He. Furthermore, we prove conjectures by Ishizeki and Nagasawa on certain parts of a decomposition of the Möbius energy and extend a theorem of Wu on inscribed polygons to curves with derivatives with vanishing mean oscillation. Finally, we show that the result by Scholtes on the -convergence of the discrete Möbius energies towards the Möbius energy also holds for curves of merely bounded energy.

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