arXiv research

Study on bending knots and energy changes in 3D space.

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…

Study on elastic curves with variable stiffness, derived from bending energy.

Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.

New solutions found for bending of flat surfaces and origami structures.

Optimal flat ribbons can be created from nonplanar curves with minimal energy.

We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we present a compactness result for branched, immersed, haunted, stratified surfac…

Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise…

Holographic principle matches deformed Liouville theory action.

Study of flat ribbons constructed along curves in 3D space.

Classifies pinned $p$-elasticae and finds unique optimality exponents.

Optimal thresholds ensure curves remain embedded in flows.

Straight lines are a basin of attraction for the elastic flow at least to level 1.9615π.

The problem of minimal distortion bending of smooth compact embedded connected Riemannian $n$-manifolds $M$ and $N$ without boundary is made precise by defining a deformation energy functional $Φ$ on the set of diffeomorphisms $\diff(M,N)$. We derive the Euler-Lagrange equation for $Φ$ and determine smooth minimizers o…

A new functional for simplicial surfaces is suggested. It is invariant with respect to Moebius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. as an application a bending energy for discrete thin-shells is derived.

Characterizes neutral deformation modes of minimal surfaces.

Symmetric elastic knots are found for certain classes with dihedral symmetry.

We preset a computational study of bending models for the curvature elasticity of lipid bilayer membranes that are relevant for simulations of vesicles and red blood cells. We compute bending energy and forces on triangulated meshes and evaluate and extend four well established schemes for their approximation: Kantor a…

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The…

Establishes a Li-Yau type inequality for curves in any codimension.

Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.

Analytic non-planar $p$-elasticae are shown to be 3D.

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…

In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.

O'Hara introduced several functionals as knot energies. One of them is the Möbius energy. We know its Möbius invariance from Doyle-Schramm's cosine formula. It is also known that the Möbius energy was decomposed into three components keeping the Möbius invariance. The first component of decomposition represents the ext…

Theory of packing diabolic domains in liquid crystals.

The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key…

This paper constructs PH spline curves with prescribed arc lengths.

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls i…

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least $C^1$ across phase boundaries. We consi…

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $Ω$. We prove existence, regularity and some structural properties of minimizers. In particular, when $Ω$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a…

Generalizes Hasimoto transformation to arbitrary flows on space curves.

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces which minimize t…

The paper classifies and analyzes the stability of elastic curves with fixed endpoints.

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional $\mathcal{W}$ is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of $\ma…

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how se…

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

Proves the bending map is proper for hyperbolic 3-manifolds.

We develop an explicit and tractable representation of a twist-grain-boundary phase of a smectic A liquid crystal. This allows us to calculate the interaction energy between grain boundaries and the relative contributions from the bending and compression deformations. We discuss the special stability of the 90 degree g…

Periodic surfaces have a limited number of bending modes, equal to their membrane modes.

Unified theory solves strain compatibility and elasticity of origami metamaterials.

Minimal surfaces can be transformed into others with unchanged bending content.

The study explores isometric deformations of surfaces of translation.

Study bends 2D surfaces in 3D space using special equations.

Study explores kinematics of surfaces under metric restrictions.