*S. Kolasinski, P. Strzelecki, and H. von der Mosel*

#### Abstract

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangentpoint 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.

#### Reference

Comm. Anal. Geom. 26 (2018), no. 6, 1251–1316