Rigidity Result Research Articles

Regularity of Conjugacies of Linearizable Generalized Interval Exchange Transformations

We consider generalized interval exchange transformations (GIETs) of d≥2 intervals which are linearizable, i.e. differentiably conjugated to standard interval exchange maps (IETs) via a diffeomorphism h of [0, 1] and study the regularity of the conjugacy h. Using a renormalization operator obtained accelerating Rauzy–Veech induction, we show that, under a full measure condition on the IET obtained by linearization, if the orbit of the GIET under renormalization converges exponentially fast in a C2 distance to the subspace of IETs, there exists an exponent 0<α<1 such that h is C1+α. Combined with the results proved by the authors in [4], this implies in particular the following improvement of the rigidity result in genus two proved in [4] (from C1 to C1+α rigidity): for almost every irreducible IET T0 with d=4 or d=5, for any GIET which is topologically conjugate to T0 via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a C1+α diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is α-Hölder continuous.

Rigidity Aspects of Penrose’s Singularity Theorem

In this paper, we study rigidity aspects of Penrose’s singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose’s singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship.

Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces: Nonexistence, mean convexity, rigidity and Calabi-Bernstein type results

Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces: Nonexistence, mean convexity, rigidity and Calabi-Bernstein type results

Analysis and Research on the crack of a type of pressed PBX column

Abstract In this paper, aiming at the crack found in a type of press loaded PBX column during ammunition charging, in order to reveal the cause of the crack, the industrial CT nondestructive testing technology was used to scan and analyze the column. The results shown that there was a bulge on the column surface in the crack area, and the internal structure of the bulge was a clearance surface parallel to the outer surface of the column; The finite element method was used to simulate the stress distribution in the pressing process of explosives and the stress after the column collides with the rigid plane. The simulation results of the pressing process shown that the stress concentration area and the area with the first plastic deformation were not at the crack position, but at the corner of the central hole; The rigid plane simulation results of column collision shown that the stress wave propagates in the axial and radial direction, which was consistent with the crack morphology and crack propagation direction on the surface of explosive column, and also matches the internal morphology of column crack. Through comprehensive analysis and judgment, the crack of the pressed PBX column was more likely to be caused by external force collision, and the probability of being caused by internal stress was very small.

A rigidity result for coisotropic submanifolds in contact geometry

A rigidity result for coisotropic submanifolds in contact geometry

On Certain Rigidity Results of Compact Regular $$(\kappa ,\mu )$$-Manifolds

On Certain Rigidity Results of Compact Regular $$(\kappa ,\mu )$$-Manifolds

Impact of printing parameters on the dynamic outputs of a slider-crank mechanism with FFF-based 3D-printed crank-arms

In this study, the crank-arm of the slider-crank mechanism was produced with different raster angles (45°, 90°, 135°) and infill densities (25%, 50%, 75%, 100%). The effect of these parameters on the dynamics of the slider was investigated experimentally and numerically, and compared with a rigid version of the crank-arm. Firstly, analytical, numerical, and experimental results are obtained for the rigid crank-arm, and the results are compared. Secondly, experiments were carried out on the same experimental setup with 12 crank-arm specimens produced using PLA. Finally, experimental results of sliders were statistically analysed with rigid-body mechanism results, and the flexibility of the crank-arms produced with fused filament fabrication was verified through numerical simulations. As the infill density decreased, the flexibility of the crank-arm increased, and the deviations in the results also grew as the energy change due to the deformation increased. The raster angle had no significant effect on the results except for 25% infill density. However, with 25% infill density, the crank-arm with 90° raster angle followed the rigid crank-arm experimental results best. As a result, it has been observed that the crank-arm produced with different printing parameters has a significant effect on the dynamics of the slider.

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

On Riemannian 4‐manifolds and their twistor spaces: A moving frame approach

AbstractIn this paper, we study the twistor space of an oriented Riemannian 4‐manifold using the moving frame approach, focusing, in particular, on the Einstein, non‐self‐dual setting. We prove that any general first‐order linear condition on the almost complex structures of forces the underlying manifold to be self‐dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first‐order quadratic conditions, showing that the Atiyah–Hitchin–Singer almost Hermitian twistor space of an Einstein 4‐manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.

Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper, we generalize this result to asymptotically hyperbolic (AH) Einstein manifold. We get an upper bound of the relative volume of AH manifold and if we combine it with the recent work of Wang and Zhou, then the rigidity is obtained.

On manifolds with nonnegative Ricci curvature and the infimum of volume growth order &lt; 2

Abstract We study open (complete and noncompact) n-manifolds M with nonnegative Ricci curvature, under the condition that any asymptotic cone of M splits off an ℝ k {\mathbb{R}^{k}} factor. In particular, we obtain two rigidity results for open n-manifolds M with Ric M ≥ 0 {\mathrm{Ric}_{M}\geq 0} and the infimum of volume growth order &lt; 2 {&lt;2} . The first result asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product ℝ × N {\mathbb{R}\times N} for some compact manifold N. The second asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n - 1 {n-1} dimensional soul.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Liouville's type results for singular anisotropic operators

Abstract We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first s s coordinate directions and of power p p , with 1 &lt; p &lt; 2 1\lt p\lt 2 along the other ( N − s ) \left(N-s) ones. First, we begin our investigation by assuming that the solution is bounded only from below, deriving a rigidity result for the range p + ( N − s ) ( p − 2 ) &gt; 0 p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

Some rigid results on doubly twisted product complex Finsler manifolds

Some rigid results on doubly twisted product complex Finsler manifolds

The Sasakian statistical structures of constant ϕ-sectional curvature on Sasakian space forms

In this paper, we investigate the Sasakian statistical structures of constant ϕ-sectional curvature based on Sasakian space forms. We obtain the classification of this kind of Sasakian statistical structures. Our classification results show that the Sasakian statistical structures of constant ϕ-sectional curvature on a Sasakian space form with dimension higher than 3 must be almost-trivial; on a 3-dimensional Sasakian space form, in addition to the almost-trivial Sasakian statistical structure, there exist other Sasakian statistical structures which satisfy the constant ϕ-sectional curvature condition. We also point out that a rigidity result for cosymplectic statistical structures of constant ϕ-sectional curvature on 3-dimensional cosymplectic space forms in [11] can be improved to the corresponding classification result.

Rigidity of the Delaunay triangulations of the plane

We prove a rigidity result for Delaunay triangulations of the plane under Luo's notion of discrete conformality, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We follow Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and develop a discrete Schwarz lemma and a discrete Liouville theorem. As a key ingredient to prove the discrete Schwarz lemma, we establish a correspondence between the Euclidean discrete conformality and the hyperbolic discrete conformality, for geodesic embeddings of triangulations. Other major tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry.

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper, we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

Soap bubbles and convex cones: optimal quantitative rigidity

We consider a class of recent rigidity results in a convex cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ \Sigma , Alexandrov’s soap bubble-type results relative to Σ \Sigma , and Heintze-Karcher’s inequality relative to Σ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ = R N \Sigma = \mathbb {R}^N . Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in ∂ Σ \partial \Sigma . We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).

Flexibility evaluation of psammophytes using Young’s modulus based on 3D numerical simulation

Flexible psammophytes play an important role in controlling soil wind erosion and desertification, owing to their characteristics. Although flexibility of psammophytes has been considered in previous studies, the interaction between flexible psammophytes and the surrounding airflow field still remained unclear. In this study, we used the Young’s modulus to describe plant flexibility and conducted a 3D computational fluid dynamics simulation using a standard k-ε model and a fluid–structure interaction model. Taking Caragana korshinskii (Caragana), a typical psammophyte, as the research object, we constructed 3D geometric models with different diameters to simulate the airflow field around the flexible psammophytes. By comparing with the simulation results of rigid plants and simulation results of flexible plants at different wind speeds, we could verify the rationality of the simulation method. Based on the simulation results, the maximum swing amplitude of the model and the Young’s modulus were found to have a negative correlation, presenting an exponential functional relationship with good fitting. The relationship between the actual Young’s modulus of the plant branches and that of different diameter models in the numerical simulation was also established. This study is expected to improve our basic understanding of the interaction between flexible psammophytes and the surrounding airflow field, and provide some qualitative reference for the numerical simulation of the airflow field around flexible psammophytes.