2015/04/17 by Sławomir Kolasiński, Kolasiński, Sławomir, Paweł Strzelecki +3 · 1 citation
Mathematics · #49J45 #49Q20 #53C21 #57R52 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Primary: 53C23 #Secondary: 49Q10 #math.AP #math.DG #math.MG #msc:49J45 #msc:49Q10 #msc:49Q20 #msc:53C21 #msc:53C23 #msc:57R52
paper · pdf · doi:10.48550/arxiv.1504.04538
44 pages, 5 figures
openalex publication_date 2015/04/17 · arxiv created 2015/10/02 · arxiv updated 2015/10/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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 ℝn. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in C1 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.