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# Communications in Analysis and Geometry

## Volume 26 (2018)

### Number 6

### Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

Pages: 1251 – 1316

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n6.a2

#### Authors

#### Abstract

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.

Received 2 October 2015

Published 29 March 2019