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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.