A Möbius invariant energy for sets of arbitrary dimension and codimension

Abstract: We consider the family of Möbius invariant energies for m-dimensional submanifolds of $\mathbb R^n$, introduced by R. Kusner and J. Sullivan, defined on a class of sets, which are given by the union of Lipschitz graphs and satisfy an additional condition of "nice" self-intersection. We show for these sets that finite energy implies Reifenberg-flatness through estimating the energy of certain subsets. This finally leads to a local representation given by a single graph and prevents any kind of self-intersection. As an immediate implication, we obtain that every immersed $C^1$ manifold with finite energy is embedded. This is joint work with Heiko von der Mosel.

Mathematics

Audience: researchers in the topic

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