Rigidity Result Research Articles

Reconstruction of groupoids and C⁎-rigidity of dynamical systems

We show how to construct a graded locally compact Hausdorff étale groupoid from a C⁎-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact étale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C⁎-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C⁎-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C⁎-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces.

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity Omega , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

Rigidity Results for Self-Shrinking Surfaces in ℝ4

Rigidity Results for Self-Shrinking Surfaces in ℝ4

Some results on almost η-Ricci-Bourguignon solitons

We determine a formula to compute the defining function of a gradient almost Ricci-Bourguignon soliton by means of the potential vector field, providing in this sense two examples. We also give a rigidity result in this case. Moreover, we deduce some curvature properties of solitons with torse-forming potential vector field. Further we generalize some of the results to almost η-Ricci-Bourguignon solitons and prove that a gradient almost Ricci-Bourguignon soliton on a doubly warped product manifold induces gradient almost η-Ricci-Bourguignon solitons on its factor manifolds.

On the fluid ball conjecture

The fluid ball conjecture states that a static perfect fluid space-time is spherically symmetric. In this paper we construct a Robinson's divergence formula for the static perfect fluid space-time. Inspired by this conjecture, a rigidity result for the spatial factor of a static perfect fluid satisfying some boundary conditions is proved, provided that an equation of state holds.

A rigidity result for normalized subfactors

We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of LΓ which is normalized by the natural copy of Γ is trivial or of finite index.

Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces. As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds.

Boundary and Rigidity of Nonsingular Bernoulli Actions

Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},\mu_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{\mu_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigidity result in the non measure preserving case. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.

From affine Poincaré inequalities to affine spectral inequalities

Given a bounded open subset Ω of Rn, we establish the weak closure of the affine ball BpA(Ω)={f∈W01,p(Ω):Epf≤1} with respect to the affine functional Epf introduced by Lutwak, Yang and Zhang in [46] as well as its compactness in Lp(Ω) for any p≥1. These points use strongly the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory of p-Rayleigh quotients in bounded domains, in the affine case, for p≥1. More specifically, we establish p-affine versions of the Poincaré inequality and some of their consequences. We introduce the affine invariant p-Laplace operator ΔpAf defining the Euler-Lagrange equation of the minimization problem of the p-affine Rayleigh quotient. We also study its first eigenvalue λ1,pA(Ω) which satisfies the corresponding affine Faber-Krahn inequality, that is, λ1,pA(Ω) is minimized (among sets of equal volume) only when Ω is an ellipsoid. This point depends fundamentally on the PDEs regularity analysis aimed at the operator ΔpAf. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for p≥1. All affine inequalities obtained are stronger and directly imply the classical ones.

On the geometry of submanifolds in certain warped products

In this paper, we deal with [Formula: see text]-dimensional submanifolds immersed in a slab of a warped product of the type [Formula: see text]. Under suitable constraints on the warping function [Formula: see text] and assuming that such a submanifold [Formula: see text] is either complete or stochastically complete, we apply some maximum principles in order to show that [Formula: see text] must be contained in a slice of [Formula: see text]. In particular, from our results we guarantee the nonexistence of [Formula: see text]-dimensional closed minimal submanifolds immersed in [Formula: see text]. Furthermore, we construct a nontrivial duo-graph in [Formula: see text] which illustrates the importance of our rigidity results.

Development and experimental validation of a mathematical model of shear rigidity of woven fabric structures

Shear is one of the most important mechanical behaviours which differentiates a woven fabric from papers and polymeric sheets. In this work a generalised mathematical model of shear rigidity of woven fabric applicable to all types of weaves has been proposed. The model demonstrates the shear rigidity of woven fabric as a function of yarn bending rigidity, fabric structural parameters and crossover firmness factor. Experimental shear rigidity results are compared with the theoretically predicted ones and a very good agreement is reached. The proposed mathematical model can able predict the shear rigidity of any woven fabric structure and thus foretells the usefulness of the fabric for a particular use.

Switchable positioning of plate-like inclusions in lipid membranes: Elastically mediated interactions of planar colloids in 2D fluids.

We demonstrate how manipulating curvature in an elastic fluid lamella enables the reversible relative positioning of flat, rigid, plate-like micrometer-scale inclusions, with spacings from about a micrometer to tens of micrometers. In an experimental model comprising giant unilamellar vesicles containing solid domain pairs coexisting in a fluid membrane, we adjusted vesicle inflation to manipulate membrane curvature and mapped the interdomain separation. A two-dimensional model of the pair potential predicts the salient experimental observations and reveals both attractions and repulsions, producing a potential minimum entirely a result of the solid domain rigidity and bending energy in the fluid membrane. The impact of vesicle inflation on domain separation in vesicles containing two solid domains was qualitatively consistent with observations in vesicles containing many domains. The behavior differs qualitatively from the pure repulsions between fluid membrane domains or interactions between nanoscopic inclusions whose repulsive or attractive character is not switchable.

Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

A Rigidity Result of Spacelike Self-Shrinkers in Pseudo-Euclidean Spaces

In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.

Initial Data Rigidity Results

We prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

Origin of hardening in spectra of cosmic ray nuclei at a few hundred GeV using AMS-02 data * *Supported by the National Natural Science Foundation of China (NSFC) (11947125, 12005124) and the Applied Basic Research Programs of Natural Science Foundation of Shanxi Province (201901D111043)

Many experiments have confirmed spectral hardening at a few hundred GeV in the spectra of cosmic ray (CR) nuclei. Three different origins have been proposed: primary source acceleration, propagation, and the superposition of different kinds of sources. In this work, a broken power law has been employed to fit each of the spectra of cosmic ray nuclei from AMS-02 directly, for rigidities greater than 45 GeV. The fitting results of the break rigidity and the spectral index differences less than and greater than the break rigidity show complicated relationships among different nuclear species, which cannot be reproduced naturally by a simple primary source scenario or a propagation scenario. However, with a natural and simple assumption, the superposition of different kinds of sources could have the potential to explain the fitting results successfully. Spectra of CR nuclei from a single future experiment, such as DAMPE, will provide us the opportunity to do cross checks and reveal the properties of the different kinds of sources.

Seismic behavior of plane frame with special-shaped CFST columns, H-shaped steel beams and vertical rib connections

The seismic behavior of special-shaped concrete-filled steel tubular (CFST) column to H-shaped steel beam plane frame (SCFSTF) with vertical rib connections was investigated experimentally and numerically in this study. Double-bay two-story SCFSTFs with the scale ratio of 1/2 were tested under constant vertical compressive load and cyclic horizontal loads (or displacements), which is a pseudo static experiment considering the axial load ratio. Based on the experimental results, failure modes, bearing capacity, hysteretic curve, skeleton curve, energy dissipation, ductility, rigidity degradation and strength degradation of SCFSTFs were investigated. As a result, a typical strong column-weak beam failure mode was determined. The degradation in strength and rigidity were minimal and energy dissipation capacity and ductility of frames were adequate, demonstrating excellent seismic performance. Besides, energy dissipation capacity and horizontal resistance were improved with axial load ratio increasing within the parametric range of this test. After the test, a finite element method based on software ABAQUS was carried out to further investigate the seismic performance. The numerical results of skeleton curve, rigidity, yield load, peak load and stress distribution agreed well with the test results. The plastic hinge formation process and ultimate failure mode were further investigated using the finite element model.

A quantitative rigidity result for a two-dimensional Frenkel–Kontorova model

We consider a Frenkel–Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina.In the discrete setting, a careful analysis of the reminders is needed to exploit these types of methodologies inspired by continuum models.

Ancient solution of mean curvature flow in space forms

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.

Puzzles and the Fatou–Shishikura injection for rational Newton maps

We establish a principle that we call the Fatou-Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (puzzle pieces). This puzzle construction is for rational Newton maps what Yoccoz puzzles are for polynomials: it provides the foundation for all kinds of rigidity results of Newton maps beyond our Fatou-Shishikura injection. Moreover, it gives necessary structure for a classification of the postcritically finite maps in the spirit of Thurston theory.