arXiv subjects

Jemal Guven

Publications and source records attributed to Jemal Guven.

At least 19 recordsLinked to original sources

arXiv2025

Equatorial deformation of homogeneous spherical fluid vesicles by a rigid ring

We examine the deformation of homogeneous spherical fluid vesicles along their equator by a circular rigid ring. We consider deformations preserving the axial and equatorial mirror symmetries of the vesicles. The configurations of the vesicle are determined employing the spontaneous curvature model subject to the constraints imposed by the ring as well as of having constant area or volume. We determine two expressions of the force exerted by the ring, one involving a discontinuity in the derivative of the curvature of the membrane across the ring, and another one in terms of the global quantities of the vesicle. For small enough values of the spontaneous curvature there is only one sequence of configurations for either fixed area or volume. The behavior of constricted vesicles is similar for both constraints, they follow a transition from prolate to dumbbell shapes, which culminates in two quasispherical vesicles connected by a small catenoid-like neck. We analyze the geometry and the force of the small neck employing a perturbative analysis about the catenoid. A stretched vesicle initially adopts an oblate shape for either constraint. If the area is fixed the vesicle increasingly flattens until it attains a disclike shape, which we examine using an asymptotic analysis. If the volume is fixed the poles approach until they touch and the vesicle adopts a discocyte shape. When the spontaneous curvature of the vesicle is close to the mean curvature of the constricted quasispherical vesicles, the sequences of configurations of both constraints develop bifurcations, and some of the configurations corresponding to one of their branches have the lowest energy.

cond-mat.soft↗

arXiv2020

Conformal Mechanics of Planar Curves

Self-similar curves arise naturally as the tension-free equilibrium states of conformally invariant bending energies. The simplest example is the Möbius invariant conformal arc-length on planar curves, dependent on the Frenet curvature $κ$ through its first derivative with respect to arc-length. There are four conserved currents associated with this invariance: the tension and torque associated with Euclidean invariance, as well as scalar and vector currents reflecting invariance under scaling and special conformal transformations respectively. If the tension vanishes, all equilibrium states are self-similar: in the case of conformal arc-length, these are logarithmic spirals with no internal structure. More generally, the tension-free states are logarithmic spirals decorated with a repeating self-similar internal structure. Here it will be shown how the conservation laws can be used to construct these curves, while also endowing their geometry with a mechanical interpretation. The scaling current and the torque together provide a scale-invariant ode for the dimensionless variable $κ'/κ^2$, which captures the internal structure of the spiral. For conformal arc-length it is constant. In tension-free states, the special conformal current vanishes. Its projections along orthogonal directions determine directly the distance from the spiral apex locally in terms of the curvature. The quadratic Casimir invariant of the Möbius group can be cast in terms of the four currents, none of which itself is invariant. For conformal arc-length, this is identified as the conformal curvature (the Schwarzian derivative of the Frenet curvature); it is constant along equilibrium curves.

nlin.SI↗

arXiv2020

Conformal symmetry breaking and self-similar spirals

Self-similar curves are a recurring motif in nature. The tension-free stationary states of conformally invariant energies describe the simplest curves of this form. Planar logarithmic spirals, for example, are associated with conformal arc-length; their unique properties reflect the symmetry and the manner of its breaking. Constructing their analogues in three-dimensions is not so simple. The qualitative behavior of these states is controlled by two parameters, the conserved scaling current $S$ and the magnitude of the torque $M.$ Their conservation determines the curvature and the torsion. If the spiral apex is located at the origin, the conserved \textit{special} conformal current vanishes. Planar logarithmic spirals occur when $M$ and $S$ are tuned so that $4MS =1$. More generally, the spiral exhibits internal structure, nutating between two fixed cones aligned along the torque axis. It expands monotonically as this pattern precesses about this axis. If the spiral is supercritical ($4MS>1$) the cones are identical and oppositely oriented. The torsion changes sign where the projection along the torque axis turns, the spiral twisting one way and then the other within each nutation. These elementary spirals provide templates for understanding a broad range of self-similar spatial spiral patterns occurring in nature. In particular, supercritical trajectories approximate rather well the nutating tip of the growing tendril in a climbing plant first described by Darwin.

cond-mat.soft↗

arXiv2020

Conformal mechanics of space curves

Any conformally invariant energy associated with a curve possesses tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations of planar logarithmic spirals. In this paper, a geometric framework is developed to construct these states explicitly using the conservation laws associated with the symmetry. The tension along a curve, conserved in equilibrium, is first constructed. While the tension itself is not invariant, the statement of its conservation is. By projecting the conservation laws along the two orthogonal invariant normal directions, the Euler-Lagrange equations are reproduced in a manifestly conformally invariant form involving the conformal curvature and torsion. The conserved torque, as well as scaling and special conformal currents implied by the symmetry are constructed explicitly. The special conformal current vanished with respect to an appropriate origin in all tension-free states. A sketch is provided of how self-similar spirals describing tension-free states can be constructed by integrating the conservation laws. The details will be provided in a companion paper, arXiv:1904.06876.

cond-mat.soft↗

arXiv2019

Pinning of Diffusional Patterns by Non-Uniform Curvature

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational symmetry. Deformations, e.g. by a bulge or indentation, break symmetry and can pin a pattern. We adapt methods of conformal mapping and perturbation theory to examine how curvature inhomogeneities select and pin patterns, and confirm the results numerically. The theory provides an analogy to quantum mechanics in a geometry-dependent potential and yields intuitive implications for cell membranes, tissues, thin films, and noise-induced quasipatterns.

cond-mat.soft↗

arXiv2019

Isometric bending requires local constraints on free edges

While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naively one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manifestly unphysical constraints are met by the boundary geometry. In this paper we identify the source of the problem: not only the local arc-length but also the geodesic curvature need to be constrained explicitly on all free edges. We derive the boundary conditions which follow. Contrary to conventional wisdom, there is no need to introduce boundary layers. This framework is applied to isolated conical defects, both with deficit as well, but more briefly, as surplus angles. Using these boundary conditions, we show that the lateral tension within a circular cone of fixed radius is equal but opposite to the radial compression, and independent of the deficit angle itself. We proceed to examine the effect of an oblique outer edge on this cone perturbatively demonstrating that both the correction to the geometry as well as the stress distribution in the cone kicks in at second order in the eccentricity of the edge.

cond-mat.soft↗

arXiv2019

Pinning of Diffusional Instabilities by Non-Uniform Curvature

Turing patterns emerge from a spatially uniform state following a linear instability driven by diffusion. Features of the eventual pattern (stabilized by non-linearities) are already present in the initial unstable modes. On a uniform flat surface or perfect sphere, the unstable modes and final patterns are degenerate, reflecting translational/rotational symmetry. This symmetry can be broken, e.g. by a bump on a flat substrate or by deforming a sphere. As the diffusion operator on a two dimensional manifold depends on the underlying curvature, the degeneracy of the initial unstable mode is similarly reduced. Different shapes can pin different modes. We adapt methods of conformal mapping and perturbation theory to analytically examine how bumps and ripples entrain modes of the diffusion operator on cylinders and spheres. We confirm these results numerically, and provide closed form expressions that describe how non-uniformities in curvature pin diffusion-driven instabilities and the resulting patterns.

cond-mat.soft↗

arXiv2017

Cylindrical confinement of semiflexible polymers

Equilibrium states of a closed semiflexible polymer binding to a cylinder are described. This may be either by confinement or by constriction. Closed completely bound states are labeled by two integers: the number of oscillations, $n$, and the number of times it winds the cylinder, $p$, the latter a topological invariant. We examine the behavior of these states as the length of the loop is increased by evaluating the energy, the conserved axial torque and the contact force. The ground state for a given $p$ is the state with $n=1$; a short loop with $p=1$ is an elliptic deformation of a parallel; as its length increases it elongates along the cylinder axis, with two hairpin ends. Excited states with $n\geq2$ and $p=1$ possess $n$-fold axial symmetry. Short (long) loops possess energies $\approx p E_0$ ($nE_0$), with $E_0$ the energy of a circular loop with same radius as the cylinder; in long loops the axial torque vanishes. Confined bound excited states are initially unstable; however, above a critical length each $n$-fold state becomes stable: the folded hairpin cannot be unfolded. The ground state for each $p$ is also initially unstable with respect to deformations rotating the loop off the surface into the interior. A closed planar elastic curve aligned along the cylinder axis making contact with the cylinder on its two sides is identified as the ground state of a confined loop. Exterior bound states behave very differently, if free to unbind, as signaled by the reversal in the sign of the contact force. If $p=1$, all such states are unstable. If $p \geq 2$, however, a topological obstruction to complete unbinding exists. If the loop is short, the bound state with $p=2$ and $n=1$ provides a stable constriction of the cylinder, partially unbinding as the length is increased. This motif could be relevant to an understanding of the process of membrane fission mediated by dynamin rings.

cond-mat.soft↗

arXiv2017

Arresting the collapse of a catenary arch

It is well known that viable architectural structures can be identified by locating the critical points of the gravitational potential energy congruent with some fixed surface metric. This is because, if the walls are thin, the lowest energy modes of deformation are strain-free, and thus described by surface isometries. If it is to stand, however, an arch had better possess some minimum rigidity. The bending energy consistent with this construction protocol, we will show, can only depend on curvature deviations away from the reference equilibrium form. The question of stability, like the determination of equilibrium, turns on the geometry. We show how to construct the self-adjoint operator controlling the response to deformations consistent with isometry. As illustration, we reassess the stability of a simple catenary arch in terms of the behavior of the ground state of this operator. The energy of this state increases monotonically with the bending rigidity and it is possible to identify the critical rigidity above which the arch is rendered stable. While this dependence may be monotonic, it exhibits a number of subcritical kinks indicating significant qualitative changes in the ground state associated with eigenvalue crossovers among the unstable modes in the spectrum; the number of such competing modes increasing rapidly as the rigidity is lowered. The initial collapse of a subcritical arch is controlled by the ground state; on the critical threshold, there are two unstable modes of equal energy, one raising the arch at its center, the other lowering it. The latter dominates as the instability grows. The qualitative behavior of the ground state changes as the rigidity is lowered---its nodal pattern as well as its parity undergoing abrupt changes as the intervals between crossovers converge---complicating the prediction of the dominant initial mode of collapse.

cond-mat.soft↗

arXiv2014

Environmental bias and elastic curves on surfaces

The behavior of an elastic curve bound to a surface will reflect the geometry of its environment. This may occur in an obvious way: the curve may deform freely along directions tangent to the surface, but not along the surface normal. However, even if the energy itself is symmetric in the curve's geodesic and normal curvatures, which control these modes, very distinct roles are played by the two. If the elastic curve binds preferentially on one side, or is itself assembled on the surface, not only would one expect the bending moduli associated with the two modes to differ, binding along specific directions, reflected in spontaneous values of these curvatures, may be favored. The shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of Lagrange multipliers to the Darboux frame associated with the curve. The forces transmitted to the surface along the surface normal will be determined. Features associated with a number of different energies, both of physical relevance and of mathematical interest, are described. The conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined.

cond-mat.soft↗

arXiv2013

Dipoles in thin sheets

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

cond-mat.soft↗

arXiv2013

Whirling skirts and rotating cones

Steady, dihedrally symmetric patterns with sharp peaks may be observed on a spinning skirt, lagging behind the material flow of the fabric. These qualitative features are captured with a minimal model of traveling waves on an inextensible, flexible, generalized-conical sheet rotating about a fixed axis. Conservation laws are used to reduce the dynamics to a quadrature describing a particle in a three-parameter family of potentials. One parameter is associated with the stress in the sheet, aNoether is the current associated with rotational invariance, and the third is a Rossby number which indicates the relative strength of Coriolis forces. Solutions are quantized by enforcing a topology appropriate to a skirt and a particular choice of dihedral symmetry. A perturbative analysis of nearly axisymmetric cones shows that Coriolis effects are essential in establishing skirt-like solutions. Fully non-linear solutions with three-fold symmetry are presented which bear a suggestive resemblance to the observed patterns.

physics.class-ph↗

arXiv2013

Constrained metric variations and emergent equilibrium surfaces

Any surface is completely characterized by a metric and a symmetric tensor satisfying the Gauss-Codazzi-Mainardi equations (GCM), which identifies the latter as its curvature. We demonstrate that physical questions relating to a surface described by any Hamiltonian involving only surface degrees of freedom can be phrased completely in terms of these tensors without explicit reference to the ambient space: the surface is an emergent entity. Lagrange multipliers are introduced to impose GCM as constraints on these variables and equations describing stationary surface states derived. The behavior of these multipliers is explored for minimal surfaces, showing how their singularities correlate with surface instabilities.

cond-mat.soft↗

arXiv2013

Force dipoles and stable local defects on fluid vesicles

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance $S$, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on $S$; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with $S$ and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.

cond-mat.soft↗

arXiv2013

Confining spheres within hyperspheres

The bending energy of any freely deformable closed surface is quadratic in its curvature. In the absence of constraints, it will be minimized when the surface adopts the form of a round sphere. If the surface is confined within a hypersurface of smaller size, however, this spherical state becomes inaccessible. A framework is introduced to describe the equilibrium states of the confined surface. It is applied to a two-dimensional surface confined within a three-dimensional hypersphere of smaller radius. If the excess surface area is small, the equilibrium states are represented by harmonic deformations of a two-sphere: the ground state is described by a quadrupole; all higher multipoles are shown to be unstable.

math-ph↗

arXiv2012

Confinement of semiflexible polymers

A variational framework is developed to examine the equilibrium states of a semi-flexible polymer that is constrained to lie on a fixed surface. As an application the confinement of a closed polymer loop of fixed length $2πR$ within a spherical cavity of smaller radius, $R_0$, is considered. It is shown that an infinite number of distinct periodic completely attached equilibrium states exist, labeled by two integers: $n=2,3,4,...$ and $p=1,2,3,...$, the number of periods of the polar and azimuthal angles respectively. Small loops oscillate about a geodesic circle: $n=2$, $p=1$ is the stable ground state; states with higher $n$ exhibit instabilities. If $R\ge 2R_0$ new states appear as oscillations about a doubly covered geodesic circle; the state $n=3, p=2$ replaces the two-fold as the ground state in a finite band of values of $R$. With increasing $R$, loop states alternate between orbital behavior as the poles are crossed and oscillatory behavior upon collapse to a multiple cover of a geodesic circle, (signalled respectively by an increase in $p$ and an increase in $n$). The force transmitted to the surface does not increase monotonically with loop size, but does asymptotically. It behaves discontinuously where $n$ changes. The contribution to energy from geodesic curvature is bounded. In large loops, the energy becomes dominated by a state independent contribution proportional to the loop size; the energy gap between the ground state and excited states disappears.

cond-mat.soft↗

arXiv2011

Conical instabilities on paper

The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a fourth-order linear differential operator. Unstretchability places a global linear constraint on these modes. Conical defects with a surplus angle exhibit an infinite number of states. If this angle is below a critical value, these states possess an n-fold symmetry labeled by an integer, n \geq 2. A nonlinear stability analysis shows that the 2-fold ground state is stable, whereas excited states possess 2(n - 2) unstable modes which come in even and odd pairs.

cond-mat.soft↗

arXiv2011

Hamiltonian dynamics for Einstein's action in G$\rightarrow$0 limit

The Hamiltonian analysis for the Einstein's action in $ G\to 0 $ limit is performed. Considering the original configuration space without involve the usual $ADM$ variables we show that the version $ Gto 0 $ for Einstein's action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.

math-ph↗