Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness and isotopy finiteness for submanifolds with uniformly
  bounded geometric curvature energies   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.

24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg