Chord arc submanifolds of arbitrary codimension

  • Bogensehnenuntermannigfaltigkeiten beliebiger Kodimension

Blatt, Simon; von der Mosel, Heiko (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2008)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2008


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.