Minimal surfaces can be transformed into others with unchanged bending content.
problem Understanding the deformation properties of minimal surfaces.
method Refined polar decomposition theorem to identify bending-neutral deformations.
result Every minimal surface can be transformed into another by a bending-neutral deformation.
Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.
problem Understanding infinitesimal deformations in branched bending complexes.
method Defining branched bending deformations, giving lower bounds, and constructing examples.
result Lower bounds on the dimension of deformation spaces and examples of specific deformations.
The study explores isometric deformations of surfaces of translation.
problem Determine the ways surfaces of translation bend isometrically.
method Analyzes existence conditions and provides closed-form expressions for infinitesimal and finite bendings of surfaces of translation.
result Surfaces of translation admit various infinitesimal and finite bendings, including purely torsional and torsion-free.
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 …
Study finds how periodic surfaces can bend without stretching.
problem Understanding isometric deformations of periodic surfaces.
method Characterization of isometric deformations using a constraint derived from Gauss theorem.
result Relates surface stretching to bending and twisting.
Periodic surfaces have a limited number of bending modes, equal to their membrane modes.
problem Understanding the limitations of bending modes in periodic surfaces.
method Analyzing deformation modes of periodic, piecewise smooth, simply connected surfaces.
result Effective membrane modes and bending modes are orthogonal, limiting the total number of modes to 3.
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…
Holographic principle matches deformed Liouville theory action.
problem Matching deformed Liouville theory action in holography.
method Developed a holographic scheme involving bending energy.
result Perfect match between deformed theory actions on field and gravity sides.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
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…
Let X be a negatively curved symmetric space and Γ a non-cocompact lattice in Isom(X). We show that small, parabolic-preserving deformations of Γ into the isometry group of any negatively curved symmetric space containing X remain discrete and faithful (the cocompact case is due to Guichard). This applie…
New solutions found for bending of flat surfaces and origami structures.
problem Understanding the energy-efficient bending modes of origami tessellations and corrugated shells.
method Direct construction of closed-form solutions for surfaces of translation.
result Three inextensional modes identified for surfaces of translation, including stretching, bending, and twisting.
Characterizes neutral deformation modes of minimal surfaces.
problem Understanding the energy content of deformation modes of minimal surfaces.
method Analyzes the energy content of stretching, drilling, and bending modes of minimal surfaces.
result All isometries of a minimal surface are globally neutral and give rise to soft elasticity.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
In [4]: `The Riley slice of Schottky space', (Proc. London Math. Soc. 69 (1994), 72-90), Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two handlebody generated by two parabolics. This theory aims to give a complete des…
New patterns deform Farey triangulation in symmetric space.
problem Deforming Farey triangulation in symmetric space.
method Realized representations of modular group as isometry groups of geodesic patterns in SL3(R)/SO(3). result 2-parameter family of deformations of Farey triangulation.
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, S3, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore…
Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
We describe a procedure to deform cubulations of hyperbolic groups by "bending hyperplanes". Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic 3-manifolds and free-by-Z groups, and Hsu-Wise turns. As an application, we show that every coco…
Paper introduces a new metric for deforming surfaces with parabolics.
problem Deformation spaces of quasifuchsian groups with parabolics.
method Developed a mapping class group invariant pressure metric on QF(S).
result Hausdorff dimension of limit sets varies analytically over QF(S).
New framework reveals limits of flexible, periodic thin surfaces.
problem Understanding the mechanical behavior of thin, periodic surfaces.
method Developed a duality between surface rotations and in-plane stresses.
result Exactly three out of six possible strain states are isometries.
Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.
problem Infimum of dual volume of convex co-compact hyperbolic 3-manifolds.
method Varying geometry by quasi-isometric deformations to deduce infimum.
result Linear lower bound on quasi-Fuchsian manifold volume based on bending lamination length.
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in PU(2,1) and using bending-connectedness. result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.
problem Transitioning between hyperbolic and Anti-de Sitter geometries.
method Construction via Half-pipe geometry on ΣimesS1 with cone singularities. result Deformation of convex core structure as bending laminations collapse.
New geometric theory explains nonuniform origami responses.
problem Understanding nonuniform responses in origami sheets.
method Purely geometric continuum theory capturing nonuniform, nonlinear response.
result Three modes govern nonuniform response, varying smoothly across the sheet.
Study explores kinematics of surfaces under metric restrictions.
problem Understanding the kinematics of surfaces under metric constraints.
method Analyzed three energy contents: stretching, drilling, and bending.
result Metric restrictions can hinder the elastic response of a shell.
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…
Study on bending knots and energy changes in 3D space.
problem Understanding energy changes in knots under small deformations.
method Analyzes infinitesimal bending of knots and energy changes using Willmore and Mobius energies.
result Changes in energy under small deformations of knots have been quantified.
Study of parabolic-preserving deformations of hyperbolic lattices.
problem Preserving parabolic subgroups during lattice deformations.
method Analysis of deformations into SU(n,1) and SO(n+1,1).
result Existence of 1-parameter families of parabolic-preserving deformations.
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
Study bends 2D surfaces in 3D space using special equations.
problem Investigate infinitesimal bendings of 2D surfaces in 3D space.
method Use Bers-Vekua type equations and systems of differential equations with periodic coefficients.
result Construct bending fields for specific classes of 2D surfaces.
Generalizes existence of bending laminations for Kleinian groups.
problem Existence of bending laminations for Kleinian surface groups.
method Generalization of Bonahon and Otal's proof to include geometrically infinite groups.
result Compactness of Kleinian groups realizing specific laminations.
The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
Study on elastic curves with variable stiffness, derived from bending energy.
problem Modeling elastic wires with varying thickness.
method Derive Euler-Lagrange equations for curves with variable bending stiffness.
result Characterizations of elastic curves with variable stiffness.
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.
This article was born as a generalisation of the analysis made by Series, where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms' Relationship proved by the author and Series to determine the asymptotic directions of pleating rays in the Mas…
Paper develops a new method to analyze 3D tree-like objects.
problem Analyzing complex geometrical and topological variations in 3D tree-like objects.
method Extended SRVF representation and new metric for tree-shaped 3D objects.
result Captures full elasticity and topological variations of branches.
Bounds projective structure norms by bending lamination lengths.
problem Bounding the L2-norm of projective structures. method Using the Thurston parameterization and Krasnov-Schlenker's W-volume theory. result Upper bounds on L2-norm of holomorphic quadratic differential by the length of bending lamination. This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
Study surface subgroups acting on projective space, finding bending laminations and spheres.
problem Surface subgroups acting on RP3 with coaffine representations. method Stratification of convex core boundary, bending laminations, and analysis of holonomy.
result Projectivization of bending data space is a sphere of dimension 6g−7. Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…
The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.
problem Understanding the role of branch points in the shape and mechanics of hyperbolic surfaces.
method Developed a discrete differential geometric (DDG) approach to study deformations of hyperbolic objects with distributed branch points.
result Branch points influence the overall morphology of hyperbolic surfaces without concentrating energy, leading to sub-exponential growth in maximum curvature.
Unified theory solves strain compatibility and elasticity of origami metamaterials.
problem Understanding and controlling the morphing paths of origami metamaterials.
method Unified theory for a wide array of origami tessellations, solving strain compatibility and elasticity.
result Origami metamaterials exhibit equal but opposite in-plane and out-of-plane Poisson's ratios and bending energy depends on strain gradient.
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…
New bounds link Schwarzian derivative to hyperbolic geometry.
problem Quantify the relationship between Schwarzian derivative and hyperbolic geometry.
method Established explicit quantitative bounds between Schwarzian norm and bending norm.
result Explicit bounds on bending norm for univalent maps with small Schwarzian norm.
This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there i…
We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy Ebend=∫κ2 together with a small multiple of ropelength R=length/thickness in order to penalize selfintersection. Our main objective is to characterize elastic…