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Abstract

We prove isotopy finiteness for various geometric curvature energies
including integral Menger curvature, and tangent-point repulsive potentials,
defined on the class of compact, embedded $m$-dimensional Lipschitz
submanifolds in ${\mathbb{R}}^n$. That is, there are only finitely many isotopy
types of such submanifolds below a given energy value, and we provide explicit
bounds on the number of isotopy types in terms of the respective energy.
Moreover, we establish $C^1$-compactness: any sequence of submanifolds with
uniformly bounded energy contains a subsequence converging in $C^1$ to a limit
submanifold with the same energy bound. In addition, we show that all geometric
curvature energies under consideration are lower semicontinuous with respect to
Hausdorff-convergence, which can be used to minimise each of these energies
within prescribed isotopy classes.