Rigidity Result Research Articles

Improvement Prediction on Contour Deformation Accuracy Using Deformable Image Registration Results Compared to Rigid Image Registration Results

The aim of this study was to analyze improvement prediction on contour deformation accuracy using deformable image registration (DIR) results compared to rigid image registration (RIR) results. Radiotherapy plans for 31 cases (seven head and neck cases, 10 chest cases, six abdomen cases and eight female pelvis cases) from the privately open database for DIR were used. These cases used at least two radiotherapy plans, and registration was performed using two plans, not only for one case but also for different cases. The DIR and RIR were performed using the DIR software MIM Maestro (MIM software Inc., Cleveland, USA). The registration results for the following organs were analyzed: eye balls, optic nerves, brain stem, spinal cord and right and left parotid glands for head and neck; right and left lungs for chest; liver and right and left kidneys for abdomen; and rectum and bladder for pelvis. Dice similarity coefficient (DSC) for the organs was calculated from the results of RIR and DIR. The improvement in the DSC was observed. DIR improved the DSC values by more than 0.2 for simple shapes, well-defined boundaries and large volumes such as eye balls, brain stem, lungs and liver. The minimum DSC for these organs was approximately 0.7. The improvement in DSC for the organs eye balls, brain stem, lungs and liver had ceiling values 0.95, 0.90, 1.0 and 1.0, respectively. DSC for the spinal cord, parotid gland, bladder and kidney also improved by DIR compared to RIR; however, DIR could not improve the DSC value for rectum compared to RIR because of a large difference in the position, shape and size due to stool and gas.

Some results about concircular vector fields on Riemannian manifolds

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs

In the first part of the article we study Hamiltonian diffeomorphisms of \mathbb R^{2n} which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo’s symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the nonlinear string equation and treat it as an infinite-dimensional Hamiltonian system. In this context we are able to apply Kuksin’s approximation by finite dimensional Hamiltonian flows and prove a PDE version of the rigidity result for coisotropic cylinders. As a particular example, this result can be applied to the sine-Gordon equation.

Rigidity properties of holomorphic Legendrian singularities

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain \Omega in a cone \Sigma in \mathbb{R}^N , N\geq 2 , and prove a rigidity result of Serrin type by showing that the existence of a solution implies that \Omega is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces \Gamma with boundary which satisfy a 'gluing' condition with respect to the cone \Sigma . We prove that if either the cone is convex or the surface is a radial graph then \Gamma must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

Bochner–Simons Formulas and the Rigidity of Biharmonic Submanifolds

New integral formulas of Simons and Bochner type are found and then used to study biharmonic and biconservative submanifolds in space forms. This leads to new rigidity results and partial answers to conjectures on biharmonic submanifolds in spheres.

A rigidity result for Kählerian manifolds endowed with closed conformal vector fields

We show that if a connected compact K¨ahlerian surface M with nonpositive Gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We also consider the case of a connected complete K¨ahlerian manifod M of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is well known that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete K¨ahlerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R2 as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of ξ.

Rigidity of Kac–Schwarz operators

In his work on the mathematical formulation of 2d quantum gravity Schwarz established a rigidity result for Kac-Schwarz operators for the n-KdV hierarchies. Later on, Adler and van Moerbeke as well as Fastr\'{e} obtained different proofs of this result. We give yet another proof of the rigidity, one that in fact holds for all Drinfeld-Sokolov hierarchies.

Volume-preserving mappings between Hermitian symmetric spaces of compact type

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian symmetric space of compact type into its Cartesian products.

Positive scalar curvature and Poincaré duality for proper actions

For G an almost-connected Lie group, we study G -equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G -invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori’s results, and an analogue of Petrie’s conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G -equivariant K -homology and K -theory, observing that Poincaré duality does not necessarily hold for general G .

Stability of Einstein Metrics on Fiber Bundles

We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian product structures on the base. As an application, we estimate the coindex of the Einstein metrics constructed in Wang and Ziller (J Differ Geom 31:215–248, 1990) and Wang (Math Z 210:305–326, 1992). Finally, we investigate more closely the linear stability of Einstein metrics from circle bundle constructions and obtain a rigidity result for linearly stable Einstein metrics of this type.

Ricci almost solitons and contact geometry

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry

Abstract The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\textrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud, and Kassel, called ${\mathbb{H}}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ${\mathbb{H}}^{2,1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$, which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$D hyperbolic geometry.

Wool fabric having thermal comfort management function via shape memory polyurethane finishing

In this study, wool fabrics were treated with shape memory polyurethane (SMPU) at different concentrations (5–20 wt%) by using pad-dry-cure process. Transition temperature (Tg) of the SMPU was suitable for body temperature so as to create a fabric having smart breathability and insulation. FT-IR and SEM analyses were conducted for assessing polymer and wool fabric interactions. Air, water vapor permeability (WVP), and water absorption capacity tests of raw and SMPU treated fabrics were carried out within temperature range covering points lower and above Tg of the polymer (20–65 °C). Effects of relative humidity (RH) on WVP were also investigated with the tests carried out under differing RH values (20–80%). According to the results, wool fabrics treated with 10 wt% SMPU had the best smart transfer capability changing according to temperature and relative humidity. Its absorption capacity was also superior with its acceptable hand values according to bending rigidity results.

Rigidity of singularities of 2D gravity water waves

We consider the Cauchy problem for the 2D gravity water wave equation. Recently Wu [27,28] proved the local well-posedness of the equation in a regime which allows interfaces with angled crests as initial data. In this work we study properties of these singular solutions and prove that the singularities of these solutions are “rigid”. More precisely we prove that an initial interface with angled crests remains angled crested, the Euler equation holds point-wise even on the boundary, the particle at the tip stays at the tip, the acceleration at the tip is the one due to gravity and the angle of the crest does not change nor does it tilt. We also show that the local well-posedness result of Wu [27,28] and our rigidity results are applicable not only to interfaces with angled crests, but also to certain types of cusps.

Ricci flow with surgery on manifolds with positive isotropic curvature

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$. In the second part, we develop a theory of ancient solutions which have bounded curvature; are $\kappa$-noncollapsed; are weakly PIC2; and are uniformly PIC. This is an extension of Perelman's work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1. In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman's surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ which do not contain non-trivial incompressible $(n-1)$-dimensional space forms.

Quantitative normal approximation of linear statistics of $\beta$-ensembles

We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.

Rigidity and exotic models for v_1-local G-equivariant stable homotopy theory

We prove that the v_1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p=2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v_1-local G-equivariant stable homotopy category, showing that for primes p ge 5 equivariant rigidity fails in general.

Maximum curvature for curves in manifolds of sectional curvature at most zero or one

We prove a sharp lower bound for the maximum curvature of a closed curve in a complete, simply connected Riemannian manifold of sectional curvature at most zero or one. When the bound is attained, we get the rigidity result. The proof utilizes the maximum principle for a suitable two-point function. In the same spirit, we also obtain a lower bound for the maximum curvature of a curve in the same ambient manifolds which has the same endpoints with a fixed geodesic segment and has a prescribed contact angle. As a corollary, the latter result applies to a curve with free boundary in geodesic balls of Euclidean space and hemisphere.

A Second Backbone: The Contribution of a Buried Asparagine Ladder to the Global and Local Stability of a Leucine-Rich Repeat Protein.

Parallel β-sheet-containing repeat proteins often display a structural motif in which conserved asparagines form a continuous ladder buried within the hydrophobic core. In such "asparagine ladders", the asparagine side-chain amides form a repetitive pattern of hydrogen bonds with neighboring main-chain NH and CO groups. Although asparagine ladders have been thought to be important for stability, there is little experimental evidence to support such speculation. Here we test the contribution of a minimal asparagine ladder from the leucine-rich repeat protein pp32 to stability and investigate lattice rigidity and hydrogen bond character using solution nuclear magnetic resonance (NMR) spectroscopy. Point substitutions of the two ladder asparagines of pp32 are strongly destabilizing and decrease the cooperativity of unfolding. The chemical shifts of the ladder side-chain HZ protons are shifted significantly downfield in the NMR spectrum and have low temperature coefficients, indicative of strong hydrogen bonding. In contrast, the HE protons are shifted upfield and have temperature coefficients close to zero, suggesting an asymmetry in hydrogen bond strength along the ladder. Ladder NH2 groups have weak 1H-15N cross-peak intensities; 1H-15N nuclear Overhauser effect and 15N CPMG experiments show this to be the result of high rigidity. Hydrogen exchange measurements demonstrate that the ladder NH2 groups exchange very slowly, with rates approaching the global exchange limit. Overall, these results show that the asparagine side chains are held in a very rigid, nondynamic structure, making a significant contribution to the overall stability. In this regard, buried asparagine ladders can be considered "second backbones" within the cores of their elongated β-sheet host proteins.