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



Sławomir Kolasiński (Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Golm, Germany)

Paweł Strzelecki (Institute of Mathematics, University of Warsaw, Poland)

Heiko von der Mosel (Institut für Mathematik, RWTH Aachen University, Aachen, Germany)


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