Rigidity Result Research Articles

Some results on higher eigenvalue optimization

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) k-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $$k\ge 3$$ . For $$k=1$$ the classical result of Weinstock (J Ration Mech Anal 3:745–753, 1954) shows that $$\sigma _1$$ is maximized by the standard metric on the round disk. For $$k=2$$ it was shown by Girouard and Polterovich (Funct Anal Appl 44(2):106–117, 2010) that $$\sigma _2$$ is not maximized for a smooth metric. We also prove a local rigidity result for the critical catenoid and the critical Mobius band as free boundary minimal surfaces in a ball under $$C^2$$ deformations. We next show that the first k Steklov eigenvalues are continuous under certain degenerations of Riemannian manifolds in any dimension. Finally we show that for $$k\ge 2$$ the supremum of the k-th Steklov eigenvalue on the annulus over all metrics is strictly larger that that over $$S^1$$ -invariant metrics. We prove this same result for metrics on the Mobius band.

Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric

We prove that a critical metric of the volume functional on a 4-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $$\mathbb {R}^{4}$$ , $$\mathbb {H}^{4}$$ or $$\mathbb {S}^{4}.$$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.

Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells

We establish Korn's interpolation inequalities and the rigidity results of the strain tensor of the middle surface for the parabolic and elliptic shells and show that the best constant in Korn's inequalities scales like $h^{3/2}$ for the parabolic shell and $h$ for the elliptic shell, removing the main assumption that the middle surface of the shell is given by one single principal coordinate in the literature and, in particular, including the closed elliptic shell.

Singular genuine rigidity

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

Comparison of sewn fabric bending rigidities obtained by heart loop method: effects of different stitch types and seam directions

Sewing quality is an important factor that contributes to the overall quality of an end-product. Sewing quality compro - mises different components such as bending, seam strength, seam slippage, elasticity etc. Among these components, bending has a special importance because of causing changes in appearance, sensorial comfort and drape of a garment. Therefore, in this study, effects of stitch type and seam direction on the bending rigidities of sewn fabrics were evaluated and compared. A polyester woven fabric which is suitable for sportswear was sewn with three basic stitch types (lock stitch, chain stitch and overlock stitch), in 5 different directions (warp, weft, 30°, 45° and 60° angles). As reference, samples without stitches were tested, too. Bending properties of samples were determined via heart loop method. According to the results, sewing increased the fabric bending rigidity. The degree of bending rigidity increment was dependent on the stitch type. Highest bending rigidity values were obtained for overlock stitched samples those were approximately 4 times higher when compared to non-sewn reference samples. Thickness of sewn parts was in accordance with the bending rigidity results. For oriented seams, bias sewing especially for 45˚ oriented samples, showed the most advantageous bending results. This study showed the usability of heart loop method for sewn samples via consistent results for different stitch types and seam directions.

Rigidity of area-minimizing $2$-spheres in $n$-manifolds with positive scalar curvature

We prove that the least area of the non-contractible immersed spheres is no more than $4\pi$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a generalization of the result in [2] to higher dimensions.

Ledoux-type rigidity results on manifolds with boundary in the presence of symmetries

We consider a 3-dimensional complete Riemannian manifold (M,g) with boundary and assuming that an optimal Sobolev or Hardy–Sobolev inequality holds with the same best constant as in the case of the solid torus T we investigate the relationship between the geometry of such a manifold and that of the solid torus and we prove Ledoux-type rigidity results (see in Ledoux (1999)) giving a positive answer to a question posed to us by the referee of our recently published paper Cotsiolis and Labropoulos (2018). Some generalization results, for n-dimensional Riemannian manifolds (M,g) with n>3, are also presented.

On the global rigidity of sphere packings on $3$-dimensional manifolds

In this paper, we prove the global rigidity of sphere packings on $3$-dimensional manifolds. This is a $3$-dimensional analogue of the rigidity theorem of Andreev–Thurston and was conjectured by Cooper and Rivin in [5]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [13].

Cayley graphs with few automorphisms

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.

Symmetry and rigidity results for composite membranes and plates

The composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.

$r$-Almost Newton-Ricci solitons immersed in a Lorentzian manifold: examples, nonexistence and rigidity

We establish the concept of $r$-almost Newton-Ricci soliton immersed into a Lorentzian manifold, which extends in a natural way the almost Ricci solitons introduced by Pigola, Rigoli, Rimoldi and Setti in [17]. In this setting, under suitable hypothesis on the potential and soliton functions, we obtain nonexistence and rigidity results. Some interesting examples of these new geometric objects are also given.

Complete spacelike hypersurfaces on symmetric spacetimes

A Lorentz manifold is said to be a conformally stationary spacetime if it is endowed with a globally defined conformal timelike vector field K, whereas it is a pp-wave when there is a globally defined parallel lightlike vector field K on M. The study of rigidity and non-existence results for spacelike hypersurfaces with constant mean curvature in these spaces has been considered in the last years by several authors. In this note we unify some known techniques and results related to both problems by considering spacelike hypersurfaces in a spacetime endowed with a globally defined conformal causal vector field. This wide family of Lorentz manifolds not only includes previous cases, but it also includes new families of spacetimes.

Hemodynamic analysis of coronary artery bypass grafting with elastic Walls and different stenosis

In this study the hemodynamic analysis of complete coronary bypass graft with elastic walls and different percentages of stenosis are investigated numerically. Blood flow is considered Newtonian and unsteady. The objective of this study is to deal with the influence of the wall elasticity, flow pulsatility and stenosis percentage on the flow configuration, Wall Shear Stress (WSS) and rotational flows. By comparing the rigid and elastic wall results of WSS, it is concluded that WSS obtains lower values in toe, heel and bed of the host vessel under the bifurcation in the elastic mode, which is closer to reality. Also it is concluded that with increasing the stenosis percentage, the possibility of occurring rotational flow will increase. The maximum and minimum values of WSS are observed in the stenosis of 70%. From the pulsatility of flow, it is observed that unsteady flow shows more accurate results also velocity and WSS have lower values compared with the steady state results.

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid–solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $$\Gamma $$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.

Invariant translators of the solvable group

We classify the translators to the mean curvature flow in the three-dimensional solvable group $$Sol_3$$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular, we show that $$Sol_3$$ admits graphical translators defined on a half plane, in contrast with a rigidity result of Shahriyari (Geom Dedicata 175:57–64, 2015) for translators in the Euclidean space. Moreover, we exhibit some nonexistence results.

The morphology of thalamic subnuclei in Parkinson's disease and the effects of machine learning on disease diagnosis and clinical evaluation

In Parkinson's disease (PD), the thalamus plays an important role in pathogenesis and disease symptoms; however, the morphological changes in thalamic subnuclei have not been clearly investigated. And there are still many challenges in individual PD diagnosis, especially clinical condition evaluations. Structural magnetic resonance imaging (MRI) was performed on 131 PD patients and 69 healthy controls (HC), and the volumes of 25 thalamic subnuclei were evaluated by FreeSurfer and a newly developed thalamus segment algorithm. Then, the individual PD diagnosis and clinical condition prediction were conducted on support vector machines (SVM) classification or regression. The bilateral thalami were enlarged; the volumes of 21 of 25 left thalamic subnuclei and 20 of 25 right thalamic subnuclei were increased, accompanied by 2 left nuclei atrophy. An accuracy of 95% with sensitivity of 97.44%, and specificity of 90.48% was achieved in PD diagnosis. United Parkinson's disease Rating Scale (UPDRS) III, limb bradykinesia, and axial akinetic symptoms score prediction were obtained with Pearson correlation coefficient of 0.5497, 0.5382, and 0.5911, respectively; however, the results of tremor, rigidity, and speech prediction were limited. Finally, accuracies of 76.92% were achieved in the UPDRS III improvement prediction. These findings confirmed that numerous left and right thalamic subnuclei were enlarged, accompanied by a few atrophies. The individual PD diagnosis, symptom, and clinical improvement prediction could be achieved based on morphology of thalamic subnuclei via machine learning.

Revisiting Leighton’s theorem with the Haar measure

AbstractLeighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

Serrin’s type overdetermined problems in convex cones

We consider overdetermined problems of Serrin’s type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.

Rigidity and unlikely intersections for formal groups

Let $K$ be a $p$-adic field and let $F$ and $G$ be two formal groups over the integers of $K$. We prove that if $F$ and $G$ have infinitely many torsion points in common, then $F=G$. This follows from a rigidity result: any bounded power series that sends