  In this paper, we establish compactness for various geometric curvature
energies including integral Menger curvature, and tangent-point repulsive
potentials, defined a priori on the class of compact, embedded $m$-dimensional
Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a
smoothing effect any sequence of submanifolds with uniformly bounded energy
contains a subsequence converging in $C^1$ to a limit submanifold.
  This result has two applications. The first one is an isotopy finiteness
theorem: 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. The second one is the lower
semicontinuity - with respect to Hausdorff-convergence of submanifolds - of all
geometric curvature energies under consideration, which can be used to minimise
each of these energies within prescribed isotopy classes.
