Chord arc submanifolds of arbitrary codimension

In this work we extend the studies of Stephen Semmes concerning hypersurfaces which satisfy a chord-arc condition with small constant to geometric objects of higher codimension and thereby we open this subject to questions arising in the field of geometric knot theory. We consider embedded, connected, and complete submanifolds of the Euclidean space without boundary which contain the point infinity. First, we show that such submanifolds satisfy a chord-arc condition with small constant and a certain Ahlfors regularity condition, if and only if the BMO-norm of the normal spaces is small and the submanifolds satisfies a Reifenberg flatness condition with a small constant. The main tool hereby is that such submanifolds contain big portions of the graph of continuous differentiable functions. Using an extension of an approximation technique due to Semmes and a new extension theorem for isotopies, we then show that these submanifolds are diffeomorphic to spheres and are unknotted.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2008

Author dc:creator

• Blatt, Simon

Contributors dc:contributor

• von der Mosel, Heiko

• info:eu-repo/classification/ddc/510
• Geometrische Analysis
• Differentialtopologie
• Knotentheorie
• Geometrische Maßtheorie
• Mathematik
• geometric Analysis
• geometric knot theory
• differential topology
• geometric measure theory

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112463%22

OAI record oai:identifier

https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A49895