On the Palais-Smale condition in geometric knot theory
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910037254864896 |
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| author | Freches, Nicolas Schumacher, Henrik Steenebrügge, Daniel von der Mosel, Heiko |
| author_facet | Freches, Nicolas Schumacher, Henrik Steenebrügge, Daniel von der Mosel, Heiko |
| contents | We prove that various families of energies relevant in
geometric knot theory satisfy the Palais-Smale condition (PS)
on submanifolds of arclength para\-metrized knots.
These energies
include linear combinations of the Euler-Bernoulli
bending energy with a wide variety of non-local
knot energies, such as
O'Hara's self-repulsive potentials $E^{α,p}$, generalized
tangent-point energies $\TP^{(p,q)}$, and generalized integral
Menger curvature functionals $\intM^{(p,q)}$. Even the
tangent-point energies $\TP^{(p,2)}$ for $p\in (4,5)$ alone
are shown to fulfill the (PS)-condition. For all energies mentioned
we can therefore prove existence of minimizing knots in any prescribed
ambient isotopy class, and we
provide long-time existence of their Hilbert-gradient flows,
and subconvergence to critical knots as time goes to infinity.
In addition, we prove $C^\infty$-smoothness of all arclength constrained
critical knots, which shows in particular that these critical knots
are also critical for the energies on the larger open set of
regular knots under a fixed-length constraint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Palais-Smale condition in geometric knot theory Freches, Nicolas Schumacher, Henrik Steenebrügge, Daniel von der Mosel, Heiko Classical Analysis and ODEs Differential Geometry Functional Analysis Geometric Topology 53C44, 35S10, 49Q10, 49N60, 57M25 We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli bending energy with a wide variety of non-local knot energies, such as O'Hara's self-repulsive potentials $E^{α,p}$, generalized tangent-point energies $\TP^{(p,q)}$, and generalized integral Menger curvature functionals $\intM^{(p,q)}$. Even the tangent-point energies $\TP^{(p,2)}$ for $p\in (4,5)$ alone are shown to fulfill the (PS)-condition. For all energies mentioned we can therefore prove existence of minimizing knots in any prescribed ambient isotopy class, and we provide long-time existence of their Hilbert-gradient flows, and subconvergence to critical knots as time goes to infinity. In addition, we prove $C^\infty$-smoothness of all arclength constrained critical knots, which shows in particular that these critical knots are also critical for the energies on the larger open set of regular knots under a fixed-length constraint. |
| title | On the Palais-Smale condition in geometric knot theory |
| topic | Classical Analysis and ODEs Differential Geometry Functional Analysis Geometric Topology 53C44, 35S10, 49Q10, 49N60, 57M25 |
| url | https://arxiv.org/abs/2505.02719 |