arXiv subjects

Cy Maor

Publications and source records attributed to Cy Maor.

At least 19 recordsLinked to original sources

arXiv2026

Energy scaling laws for thin elastic sheets with topological defects

We derive energy scaling laws for thin elastic sheets with topological defects --- disclinations and dislocations --- for a fully nonlinear 3D model. For disclinations, the scaling laws are tight in the thickness parameter, and improve upon previous results by applying simultaneously to positive and negative disclinations (e-cones) and by giving an explicit dependence on the defect parameter; this latter dependence is, however, still not tight. For thin bodies with dislocations, these are, to the best of our knowledge, the first rigorous bounds for models of finite thickness, are tight when the Burgers vector is not large with respect to the thickness, and relate to a well-known conjecture from the physics literature about the scaling. A main tool is modeling these bodies in the framework of non-Euclidean elasticity, as bodies with a curl-free pre-strain; the curl-freeness allows us to obtain geometric rigidity estimates for the lower bounds.

math.AP↗

arXiv2026

Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces

The space of $W^{2,2}$-isometric immersions of a surface into $\mathbb{R}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy --- the $L^2$-norm of the second fundamental form --- is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where "too many" asymptotic directions meet --- or, equivalently, where the index of the Gauss map is not $-1$ --- as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$-isometric immersion. We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$-convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$-isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$-isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$-isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.

math.DG↗

arXiv2025

Curvature Potential Formulation for Thin Elastic Sheets

Thin elastic sheets appear in systems ranging from graphene to biological membranes, where phenomena such as wrinkling, folding, and thermal fluctuations originate from geometric nonlinearities. These effects are treated within weakly nonlinear theories, such as the Foppl-von Karman equations, which require small slopes and fail when deflections become large even if strains remain small. We introduce a methodological progress via a geometric reformulation of thin-sheet elasticity based on a stress potential and a curvature potential. This formulation preserves the structure of the classical equations while extending their validity to nonlinear, multivalued configurations, and geometrically frustrated states. The framework provides a unified description of thin-sheet mechanics in regimes inaccessible to existing theories and opens new possibilities for the study of elastic membranes and two-dimensional materials.

cond-mat.soft↗

arXiv2025

Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces. We expect that this approach will be useful for obtaining completeness results for other manifolds of mappings.

math.DG↗

arXiv2025

The Willmore energy and curvature concentration

We study isometric immersions of a Riemannian surface $(\Omega,\frak{g})$, where $\Omega \subset \mathbb{R}^2$, into $\mathbb{R}^3$. We consider their bending energy, i.e., the square of the $L^2$-norm of their second fundamental form, which is equivalent to the Willmore functional. We obtain two new lower bounds for this energy, one in terms of the Gaussian curvature of the surface, and the other in terms of a Burgers vector -- a measure of non-flatness connected to torsion. These new estimates provide optimal blowup rates of the energy when the curvature is concentrated (e.g., in a conical geometry). In the more subtle case of dipoles of concentrated curvature, we use the Burgers vector estimates to obtain an optimal blowup rate in terms of the size of the system. Our motivation comes from non-Euclidean elasticity, in which cones and curvature-dipoles play a central role. The lower bounds derived in this work directly yield lower bounds for the elastic energy of thin elastic sheets. The derivation of the curvature-based lower bound involves an isoperimetric inequality for framed loops, which we believe to be of independent interest.

math.DG↗

arXiv2025

A Riemannian viewpoint on the Amari-Cencov $\alpha$-connections and Proudman-Johnson equations

We give a new geometric interpretation of the Amari-Cencov $\alpha$-connections $\nabla^{(\alpha)}$ from information geometry: On the space of densities $\operatorname{Dens}_+(M)$, we show that there exist Riemannian metrics $G^\alpha$, which we call $\alpha$-Fisher-Rao metrics, whose Levi-Civita connections are $\nabla^{(\alpha)}$. With the exception of $\alpha=0$ (the Fisher-Rao metric), these metrics are non-invariant to the action of the diffeomorphism group $\operatorname{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\nabla^{(\alpha)}$ as energy-minimizing curves. On the space of probability densities $\operatorname{Prob}(M)$, we show that the same phenomenon holds for $\alpha\in \{-1,0,1\}$ and that the $\alpha$-connections are not metric otherwise. We show that $\nabla^{(\alpha)}$-geodesics on this space can be interpreted as radial projections of straight lines on appropriate hyper-surfaces, and use this geometric picture to obtain geodesic convexity for any $\alpha\in \mathbb{R}$. In addition, we prove analogous results for appropriate metrics and connections on $\operatorname{Diff}(M)$, which, for the case $M=\mathbb{R}$, imply that the generalized Proudman-Johnson equations on the real line are the Euler-Arnold equations of non-right invariant metrics. Finally, in the finite-dimensional case, we show that $\nabla^{(\alpha)}$ can be metric or non-metric depending on the considered statistical model.

math.DG↗

arXiv2025

Rigorous analysis of shape transitions in frustrated elastic ribbons

Ribbons are elastic bodies of thickness $t$ and width $w$ with $t\ll w\ll 1$ (after appropriate nondimensionalization). Many ribbons in nature have a non-trivial internal geometry, making them incompatible with Euclidean space. This incompatibility -- expressed mathematically as a failure of the Gauss-Codazzi equations for surfaces -- can trigger shape transitions between narrow and wide ribbons. These transitions depend on the internal geometry: ribbons whose incompatibility arises from failure of the Gauss equation always exhibit a transition, whereas those whose incompatibility arises from failure of the Codazzi equations, may or may not. We give the first rigorous analysis of this phenomenon, mainly for ribbons whose first fundamental form is flat. For Gauss-incompatible ribbons we identify the natural energy scaling of the problem and prove the existence of a shape transition. For Codazzi-incompatible ribbons we give a necessary condition for a transition to occur. Furthermore, our study reveals a fundamental distinction: the transition is "microscopic" for Gauss-incompatible ribbons, persisting as the width tends to $0$, whereas it is "mesoscopic" for Codazzi-incompatible ribbons, observable only at small but finite width. The results are obtained by calculating the $\Gamma$-limits, as $t,w\to 0$, for narrow ribbons ($w^2 \ll t$), and wide ribbons (taking $t$ to zero and then $w$), in the natural energy scalings dictated by the internal geometry.

math.AP↗

arXiv2024

On material-uniform elastic bodies with disclinations and their homogenization

In this note, we define material-uniform hyperelastic bodies (in the sense of Noll) containing discrete disclinations and dislocations, and study their properties. We show in a rigorous way that the size of a disclination is limited by the symmetries of the constitutive relation; in particular, if the symmetry group of the body is discrete, it cannot admit arbitrarily small, yet non-zero, disclinations. We then discuss the application of these observations to the derivations of models of bodies with continuously-distributed defects.

math-ph↗

arXiv2024

A continuum geometric approach for inverse design of origami structures

Miura-Ori, a celebrated origami pattern that facilitates functionality in matter, has found multiple applications in the field of mechanical metamaterials. Modifications of Miura-Ori pattern can produce curved configurations during folding, thereby enhancing its potential functionalities. Thus, a key challenge in designing generalized Miura-Ori structures is to tailor their folding patterns to achieve desired geometries. In this work, we address this inverse-design problem by developing a new continuum framework for the differential geometry of generalized Miura-Ori. By assuming that the perturbation to the classical Miura-Ori is slowly varying in space, we derive analytical relations between geometrical properties and the perturbation field. These relationships are shown to be invertible, allowing us to design complex curved geometries. Our framework enables porting knowledge, methods and tools from continuum theories of matter and differential geometry to the field of origami metamaterials.

cond-mat.soft↗

arXiv2024

Linearization in incompatible elasticity for general ambient spaces

Motivated by recent interest in elastic problems in which the target space is non-Euclidean, we study a limit where local rest distances within an elastic body are incompatible, yet close to, distances within the ambient space. Specifically, we obtain, via $\Gamma$-convergence, a limit elastic model for a sequence of elastic bodies $(M,g_\varepsilon)$ in an ambient space $(S,s)$, for Riemannian metrics $g_\varepsilon$ and $s$ such that $g_\varepsilon \to s$. Furthermore, we relate the minimum of the limit problem to a linearized curvature discrepancy between $g_\varepsilon$ and $s$, using recent results of Kupferman and Leder. This relation confirms a linearized version of a long-standing conjecture in elasticity regarding the relation between the elastic energy and the curvature of the underlying space. The main technical challenge, compared to other linearization results in elasticity, is obtaining the correct notion of displacement for manifold-valued configurations, using Sobolev truncations and parallel transport. We show that the associated compactness result is obtained if $(S,s)$ satisfies a quantitative rigidity property, analogous to the Friesecke--James--M\"uller rigidity estimate in Euclidean space, and show that this property holds when $(S,s)$ is a round sphere.

math.AP↗

arXiv2023

Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional order $q\in [0,\infty)$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if $q>1/2$. Our second main result shows that the metric is geodesically-complete (i.e., the geodesic equation is globally well-posed) if $q>3/2$, whereas if $q<3/2$ then finite-time blowup may occur. The geodesic-completeness for $q>3/2$ is obtained by proving metric-completeness of the space of $H^q$-immersed curves with the distance induced by the Riemannian metric.

math.DG↗

arXiv2023

The $L^p$-Fisher-Rao metric and Amari-Cencov $\alpha$-connections

We introduce a family of Finsler metrics, called the $L^p$-Fisher-Rao metrics $F_p$, for $p\in (1,\infty)$, which generalizes the classical Fisher-Rao metric $F_2$, both on the space of densities Dens$_+(M)$ and probability densities Prob$(M)$. We then study their relations to the Amari-\u{C}encov $\alpha$-connections $\nabla^{(\alpha)}$ from information geometry: on Dens$_+(M)$, the geodesic equations of $F_p$ and $\nabla^{(\alpha)}$ coincide, for $p = 2/(1-\alpha)$. Both are pullbacks of canonical constructions on $L^p(M)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of $\alpha$-geodesics as being energy minimizing curves. On Prob$(M)$, the $F_p$ and $\nabla^{(\alpha)}$ geodesics can still be thought as pullbacks of natural operations on the unit sphere in $L^p(M)$, but in this case they no longer coincide unless $p=2$. Using this transformation, we solve the geodesic equation of the $\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of $F_p$, and study their relation to $\nabla^{(\alpha)}$.

math.DG↗

arXiv2023

Stability of isometric immersions of hypersurfaces

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold $M^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:M^d\to N^{d+1}$, whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result by the authors to a general target manifold $N$, removing a constant curvature assumption. The method of proof differs from that in Alpern et al.: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to Ciarlet et al. (Anal. Appl. 2019) but with no a-priori assumed bounds.

math.DG↗

arXiv2023

From Volterra dislocations to strain-gradient plasticity

We rigorously derive a strain-gradient model of plasticity as a $\Gamma$-limit of continuum bodies containing finitely-many edge-dislocations (in two dimensions). The key difference from previous such derivations is the elemental notion of a dislocation: we work in a continuum framework in which the lattice structure is represented by a smooth frame field, and the presence of a dislocation manifests in a circulation condition on that frame field; the resulting model is a Lagrangian approach with a multiplicative strain decomposition. The multiplicative nature of the geometric incompatibility generates many technical challenges, which require a systematic study of the geometry of bodies containing multiple dislocations, the definition of new notions of convergence, and the derivation of new geometric rigidity estimates pertinent to dislocated bodies. Our approach places the strain-gradient limit in a unified framework with other models of dislocations, which cannot be addressed within the "admissible strain" approach used in previous works.

math.AP↗

arXiv2021

Hierarchy of Geometrical Frustration in Elastic Ribbons: shape-transitions and energy scaling obtained from a general asymptotic theory

Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configuration depending on the relation between their width and thickness. We show that the existence of such a transition, and the scaling at which it occurs, strongly depend on the system considered. Using an asymptotic approach, treating the width as a small parameter, we find the leading energy terms resulting from the frustration and predict the existence and scaling of the shape transition. We study in detail 5 different types of frustrated ribbons with a different morphological dependence on ribbon's width: a sharp shape-transition at a critical width, a moderate transition with an intermediate regime, and no transition at all. We show that the predictions of our approach match experimental results from two different experimental systems: prestressed rubber bilayers and 4D printed thermoplastics, in a wide variety of geometric settings.

cond-mat.soft↗

arXiv2021

A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations

We show that two families of equations on the real line, the generalized inviscid Proudman--Johnson equation, and the $r$-Hunter--Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman--Johnson equations as geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on an appropriate diffeomorphism group on $\mathbb{R}$. Generalizing a construction of Lenells for the Hunter--Saxton equation, we analyze the $r$-Hunter--Saxton equation using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equation on the $L^r$-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.

math.DG↗

arXiv2020

Asymptotic rigidity for shells in non-Euclidean elasticity

We consider a prototypical "stretching plus bending" functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.

math.DG↗

arXiv2020

Reference configurations vs. optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full $\Gamma$-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The $\Gamma$-limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additional term is zero and the limit energy reduces to standard linear elasticity.

math.AP↗