Heiko von der Mosel

Integral Menger curvature for surfaces

Abstract We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C 1 , λ -a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R > 0 which depends only on an upper bound E for the integral Menger curvature M p ( Σ ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) M p -energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Holder exponent λ up to the optimal one, λ = 1 − ( 8 / p ) , thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.

Tangent-point self-avoidance energies for curves

We study a two-point self-avoidance energy which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the 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 for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q > 2 the energy evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite -energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent qcrit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Holder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

Energetics and dynamics of global integrals modeling interaction between stiff filaments

The attractive and spacing interaction between pairs of filaments via cross-linkers, e.g. myosin oligomers connecting actin filaments, is modeled by global integral kernels for negative binding energies between two intersecting stiff and long rods in a (projected) two-dimensional situation, for simplicity. Whereas maxima of the global energy functional represent intersection angles of ‘minimal contact’ between the filaments, minima are approached for energy values tending to −∞, representing the two degenerate states of parallel and anti-parallel filament alignment. Standard differential equations of negative gradient flow for such energy functionals show convergence of solutions to one of these degenerate equilibria in finite time, thus called ‘super-stable’ states. By considering energy variations under virtual rotation or translation of one filament with respect to the other, integral kernels for the resulting local forces parallel and orthogonal to the filament are obtained. For the special modeling situation that these variations only activate ‘spring forces’ in direction of the cross-links, explicit formulas for total torque and translational forces are given and calculated for typical examples. Again, the two degenerate alignment states are locally ‘super-stable’ equilibria of the assumed over-damped dynamics, but also other stable states of orthogonal arrangement and different asymptotic behavior can occur. These phenomena become apparent if stochastic perturbations of the local force kernels are implemented as additive Gaussian noise induced by the cross-link binding process with appropriate scaling. Then global filament dynamics is described by a certain type of degenerate stochastic differential equations yielding asymptotic stationary processes around the alignment states, which have generalized, namely bimodal Gaussian distributions. Moreover, stochastic simulations reveal characteristic sliding behavior as it is observed for myosin-mediated interaction between actin filaments. Finally, the forgoing explicit and asymptotic analysis as well as numerical simulations are extended to the more realistic modeling situation with filaments of finite length.

Conformal mapping of multiply connected Riemann domains by a variational approach

Abstract We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in ℝ2 can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebes mapping theorem.

On some knot energies involving Menger curvature

Abstract 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.

On sphere-filling ropes

Abstract What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the ropes thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seam lines 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.

Elastic knots in euclidean 3-space

Abstract This paper deals with the problem of minimizing the curvature functional f κ2 ds on isotopy classes of closed knotted curves in R 3. We show existence of minimizers under a given topological knot type and develop a regularity theory by analyzing different touching situations.

The Partially Free Boundary Problem for Parametric Double Integrals

We prove the existence of conformally paramaterized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class \( H_{loc}^{2,2} \cap C^{1,\alpha } ,0 < \alpha < 1 \), up to the free boundary under the assumption that there exists a perfect dominance function in the sense of Morrey.

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.

Dominance Functions for Parametric Lagrangians

We discuss the concept of dominance functions for parametric Lagrangians together with important examples and various applications to the existence and regularity theory for minimizers of parametric functionals, and for the construction of unstable stationary surfaces. The focus lies on the construction of a perfect dominance function based on ideas of Morrey.