No. 2018.08
Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies
S. Kolasinski, P. Strzelecki, and H. von der Mosel
Subject: Menger curvature, tangent-point energies, compactness, semicontinuity, isotopy finiteness, topological constraints


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.


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