arXiv subjects

Heiko von der Mosel

Publications and source records attributed to Heiko von der Mosel.

At least 19 recordsLinked to original sources

arXiv2026

Elastic BB knots are multifold circles

We study elastic knots obtained by minimizing the bending energy of a unit loop together with a small multiple of ropelength. We assume a generalized Fáry-Milnor inequality on the $C^1$-boundary of a knot class: every $H^2$-limit curve in the $C^1$-closure has total curvature at least $2π$ times the bridge index. Under this assumption we prove that every elastic knot in a one-component BB knot class, that is, a class whose bridge and braid indices coincide, is the round circle covered precisely that many times. The proof does not differentiate the nonsmooth ropelength functional. A minimal closed braid is contracted in the fibres of a solid torus, producing a family of comparison curves with quadratic bending-energy excess and thickness of linear order. This yields a lower thickness bound for the regularized minimizers. Variations on an intermediate scale then give free stationarity of the limiting curve for the normalized bending energy. Constant curvature and free stationarity reduce the classification to a linear distributional ordinary differential equation.

math.DG↗

arXiv2026

On the Palais-Smale condition in geometric knot theory

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.

math.CA↗

arXiv2025

Stability of knot equivalence at low regularity, and symmetric critical knots for the Möbius energy

We present sufficient criteria for the equivalence of tame knots at low regularity. To this end, we introduce a localized version of Gromov's distortion for any closed path-connected subset of $\R^n$. If two such sets have local Gromov distortion below a universal dimension-dependent constant $g_n$ at some scale, and if their Hausdorff-distance is less than one quarter of that scale, we can show that the fundamental groups of their complements are isomorphic. In addition, we construct this isomorphism so that it restricts to the corresponding peripheral subgroups as an isomorphism as well. Applied to the images of one-dimensional knots it follows that two knots are equivalent if their Hausdorff-distance is bounded in terms of the scale under which their local Gromov distortion is controlled. From that we deduce novel stability results for knot equivalence in the Lipschitz category, and in the setting of fractional Sobolev regularity below $C^1$. Moreover, we prove a compactness theorem of knot equivalence classes with respect to weak $W^{3/2,2}$-convergence. As an application we show that the Möbius energy introduced by O'Hara~\cite{ohara_1991a} can be minimized within arbitrary prime knot classes under a symmetry constraint, and that these minimizers are in fact critical points and therefore smooth and even real analytic. In particular, in every torus knot class there are at least two distinct critical knots for the Möbius energy.

math.GT↗

arXiv2025

Optimal planar immersions of prescribed winding number and Arnold invariants

Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter $δ > 0$ in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as ${δ\to 0}$. Moreover, we show that any sequence of minimizers subconverges in ${C^1}$, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter $δ$ is sufficiently small. Therefore, this limit curve serves as an "optimal" curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.

math.CA↗

arXiv2022

Banach gradient flows for various families of knot energies

We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O'Hara's self-repulsive potentials $E^{α,p}$. In order to do so, we employ the theory of curves of maximal slope in slightly smaller spaces compactly embedding into the respective energy spaces associated to these functionals, and add a term involving the logarithmic strain, which controls the parametrisations of the flowing (knotted) loops. As a prerequisite, we prove in addition that O'Hara's knot energies $E^{α,p}$ are continuously differentiable.

math.CA↗

arXiv2022

Tangent-point energies and ropelength as Gamma-limit of discrete tangent-point energies on biarc curves

Using interpolation with biarc curves we prove $Γ$-convergence of discretized tangent-point energies to the continuous tangent-point energies in the $C^1$-topology, as well as to the ropelength functional. As a consequence discrete almost minimizing biarc curves converge to ropelength minimizers, and to minimizers of the continuous tangent-point energies. In addition, taking point-tangent data from a given $C^{1,1}$-curve $γ$, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of $γ$, together with an explicit convergence rate.

math.CA↗

arXiv2021

Symmetric elastic knots

Minimizing the bending energy within knot classes leads to the concept of elastic knots which has been initiated in [von der Mosel, Asymptot. Anal. 1998]. Motivated by numerical experiments in arxiv:1804.02206 (doi:10.1090/mcom/3633) we prescribe dihedral symmetry and establish existence of dihedrally symmetric elastic knots for knot classes admitting this type of symmetry. Among other results we prove that the dihedral elastic trefoil is the union of two circles that form a (planar) figure-eight. We also discuss some generalizations and limitations regarding other symmetries and knot classes.

math.DG↗

arXiv2021

A speed preserving Hilbert gradient flow for generalized integral Menger curvature

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case, and provide $C^{1,1}$-bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical $L^2$ gradient descent or other optimization methods.

math.CA↗

arXiv2021

M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions

In the present paper we investigate generalizations of O'Hara's Möbius energy on curves \cite{ohara_1991a}, to Möbius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set $Σ\subset \R^n$ that locally finite energy implies that $Σ$ is, in fact, an embedded Lipschitz submanifold of $\R^n$ -- sometimes even smoother (depending on the a priorily given additional regularity of the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set $Σ$ is already sufficient to guarantee finite energy of $Σ$. This type of Sobolev regularity is exactly what one would expect in view of Blatt's characterization \cite{blatt_2012a} of the correct energy space for the Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy $E_\textnormal{KS}$ \cite{kusner-sullivan_1997} since one of the energies considered here is equivalent to $E_\textnormal{KS}$.

math.DG↗

arXiv2020

Symmetric critical knots for O'Hara's energies

We prove the existence of symmetric critical torus knots for O'Hara's knot energy family $E_α$, $α\in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth $E_α$-critical knots, which supports experimental observations using numerical gradient flows.

math.CA↗

arXiv2019

Singular Support of Minimizers of the Causal Variational Principle on the Sphere

The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $τ>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $τ>\sqrt{6}$, the support is proven to have Hausdorff dimension at most $6/7$.

math.CA↗

arXiv2016

The elastic trefoil is the twice covered circle

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic knots, i.e., all limit configurations of energy minimizers of the total energy $E_\vartheta:=E_{\text{bend}}+\vartheta\mathcal R$ as $\vartheta$ tends to zero. The elastic unknot turns out to be the round circle with bending energy $(2π)^2$. For all (non-trivial) knot classes for which the natural lower bound $(4π)^2$ for the bending energy is sharp, the respective elastic knot is the twice covered circle. The knot classes for which $(4π)^2$ is sharp are precisely the $(2,b)$-torus knots for odd $b$ with $|b|\ge 3$ (containing the trefoil). In particular, the elastic trefoil is the twice covered circle.

math.DG↗

arXiv2015

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.

math.DG↗

arXiv2014

On minimal immersions in Finsler spaces

We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we establish the existence of smooth Finsler-minimal immersions spanning given extreme or graphlike boundary contours.

math.DG↗

arXiv2012

On some knot energies involving Menger curvature

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.

math.CA↗

arXiv2012

Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $Σ^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $Σ$ satisfying a mild general condition relating the size of holes in $Σ$ to the flatness of $Σ$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,τ}\cap W^{2,p}$, where $p>m$ and $τ=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $Σ$ or (b) the size of spheres tangent to $Σ$ at one point and passing through another point of $Σ$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(Σ)$ for $p>m$ if and only if the local graph representations of $Σ$ have second order derivatives in $L^p$ and $Σ$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $Σ$ is a round sphere.

math.CA↗

arXiv2012

Plateau's problem in Finsler 3-space

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.

math.DG↗

arXiv2011

Tangent-point repulsive potentials for a class of non-smooth $m$-dimensional sets in $\R^n$. Part I: Smoothing and self-avoidance effects

We consider repulsive potential energies $\E_q(Σ)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $Σ$ in $\R^n.$ Finiteness of the energy $\E_q(Σ)$ has three sorts of effects for the set $Σ$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $Σ$ onto suitable $m$-planes and therefore large $m$-dimensional Hausdorff measure of $Σ$ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey-Sobolev embedding theorem: Any admissible set $Σ$ with finite $\E_q$-energy, for any exponent $q>2m$, is, in fact, a $C^1$-manifold whose tangent planes vary in a Hölder continuous manner with the optimal Hölder exponent $μ=1-(2m)/q$. Moreover, the patch size of the local $C^{1,μ}$-graph representations is uniformly controlled from below only in terms of the energy value $\E_q(Σ)$.

math.CA↗