31
31

Sep 23, 2013
09/13

by
Pawel Strzelecki; Heiko von der Mosel

texts

#
eye 31

#
favorite 0

#
comment 0

We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no...

Source: http://arxiv.org/abs/1006.4566v2

46
46

Sep 18, 2013
09/13

by
Paweł Strzelecki; Marta Szumańska; Heiko von der Mosel

texts

#
eye 46

#
favorite 0

#
comment 0

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...

Source: http://arxiv.org/abs/1209.1527v2

3
3.0

Jun 30, 2018
06/18

by
Patrick Overath; Heiko von der Mosel

texts

#
eye 3

#
favorite 0

#
comment 0

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.

Topics: Mathematics, Analysis of PDEs, Differential Geometry, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1403.8025

3
3.0

Jun 28, 2018
06/18

by
Henryk Gerlach; Philipp Reiter; Heiko von der Mosel

texts

#
eye 3

#
favorite 0

#
comment 0

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\int\kappa^2$ 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...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1510.06171

55
55

Jul 20, 2013
07/13

by
Sławomir Kolasiński; Paweł Strzelecki; Heiko von der Mosel

texts

#
eye 55

#
favorite 0

#
comment 0

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The...

Source: http://arxiv.org/abs/1203.4688v2

70
70

Sep 18, 2013
09/13

by
Pawel Strzelecki; Heiko von der Mosel

texts

#
eye 70

#
favorite 0

#
comment 0

We consider repulsive potential energies $\E_q(\Sigma)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in $\R^n.$ Finiteness of the energy $\E_q(\Sigma)$ has three sorts of effects for the set $\Sigma$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $\Sigma$ onto suitable $m$-planes and therefore large $m$-dimensional...

Source: http://arxiv.org/abs/1102.3642v1

15
15

Jun 27, 2018
06/18

by
Sławomir Kolasiński; Paweł Strzelecki; Heiko von der Mosel

texts

#
eye 15

#
favorite 0

#
comment 0

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...

Topics: Differential Geometry, Analysis of PDEs, Mathematics, Metric Geometry

Source: http://arxiv.org/abs/1504.04538

105
105

Jul 19, 2013
07/13

by
Henryk Gerlach; Heiko von der Mosel

texts

#
eye 105

#
favorite 0

#
comment 0

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.

Source: http://arxiv.org/abs/1005.4609v1

37
37

Sep 18, 2013
09/13

by
Patrick Overath; Heiko von der Mosel

texts

#
eye 37

#
favorite 0

#
comment 0

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.

Source: http://arxiv.org/abs/1209.2327v1