arXiv research

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

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

New bounds link Schwarzian derivative to hyperbolic geometry.

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

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

Let $S$ be any closed hyperbolic surface and let $λ$ be a maximal geodesic lamination on $S$. The amount of bending of an abstract pleated surface (homeomorphic to $S$) with the pleating locus $λ$ is completely determined by an $(\mathbb{R}/2π\mathbb{Z})$-valued finitely additive transverse cocycle $β$ to the geodesic …

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

The study explores isometric deformations of surfaces of translation.

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

We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence …

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…

Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.

Grafting is a method of obtaining new projective structures from a hyperbolic structure, basically by gluing a flat cylinder into a surface along a closed geodesic in the hyperbolic structure, or by limits of that procedure. This induces a map of Teichmuller space to itself. We prove that this map is a homeomorphism by…

Generalizes existence of bending laminations for Kleinian groups.

The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.

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

We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.

Generalizes Hasimoto transformation to arbitrary flows on space curves.

Study finds how periodic surfaces can bend without stretching.

Bounds projective structure norms by bending lamination lengths.

This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.

Study surface subgroups acting on projective space, finding bending laminations and spheres.

Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.

Proposes a new network to improve nuclei segmentation in histopathology images.

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…

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

Twisted $\operatorname{SL}_2 \mathbb{C}$ local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …

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

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…

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.

Let $M$ and $N$ be compact smooth oriented Riemannian $n$-manifolds without boundary embedded in $\mathbb{R}^{n+1}$. Several problems about minimal distortion bending and morphing of $M$ to $N$ are posed. Cost functionals that measure distortion due to stretching or bending produced by a diffeomorphism $h:M \to N$ are …

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, $\mathbb{S}^3$, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore…

Graph Prolongation Convolutional Networks improve model performance in microtubule bending simulations.

Critical trajectories in a sphere are found for a specific bending functional.

Constructs minimal surfaces by gluing saddle towers with Scherk ends.

A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case o…

Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of $f$ by…

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map which associates to each structure in P(N,C) the lengths of the curves in the b…

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…

Study complex hyperbolic structures on a disc orbibundle with 5 cone points.

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic…

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…

Paper introduces a new metric for deforming surfaces with parabolics.

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimate…

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

Holographic principle matches deformed Liouville theory action.