Boundary Rigidity Research Articles

The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity

Abstract We consider an MP -system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple MP -systems, we study the MP -ray transform in order to obtain new boundary rigidity results for MP -systems. We show that there is an explicit relation between the MP -ray transform and the magnetic one, which allows us to apply results from Dairbekov et al (2007 Adv. Math. 216 535–609) to our case. Regarding rigidity, we show that there exists a generic set G m of simple MP -systems, which is open and dense, such that any two MP -systems close to an element in it and having the same boundary action function, must be k-gauge equivalent.

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g) is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric dg induced on the boundary ∂M? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in R3 can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

Filling volume minimality and boundary rigidity of metrics close to a negatively curved symmetric metric

Filling volume minimality and boundary rigidity of metrics close to a negatively curved symmetric metric

Boundary Rigidity, and Non-Rigidity, of Projective Structures

Boundary Rigidity, and Non-Rigidity, of Projective Structures

Marked boundary rigidity for surfaces of Anosov type

Marked boundary rigidity for surfaces of Anosov type

A note on the n$n$‐point boundary rigidity

AbstractIn this note, we prove some general ‐point boundary rigidity theorems which can be applied to various classes of bounded taut domains.

A new Schwarz-Pick lemma at the boundary and rigidity of holomorphic maps

In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in CN in the spirit of the boundary Schwarz lemma of Burns–Krantz. Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz. The proof is based on a new Harnack–type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained. As an auxiliary tool we establish a Hurwitz–type result about preservation of zeros of sequences of conformal pseudometrics. Thirdly, we apply the one–dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz–Pick lemma for holomorphic maps of strongly convex domains in CN for N>1.

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

The Burns-Krantz rigidity with an interior fixed point

We prove a Burns-Krantz type boundary rigidity near strongly pseudoconvex points for holomorphic self-maps with an interior fixed point. This confirms a conjecture of Huang.

Filling Volume Minimality and Boundary Rigidity of Metrics Close to a Negatively Curved Symmetric Metric

Filling Volume Minimality and Boundary Rigidity of Metrics Close to a Negatively Curved Symmetric Metric

Non-tangential Burns-Krantz Rigidity

We prove some non-tangential Burns-Krantz type boundary rigidity theorems.

The Nehari-Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions

"We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self{maps of the unit disk. In particular, we discuss the case of in nitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary."

Two boundary rigidity results for holomorphic maps

Two boundary rigidity results for holomorphic maps

Boundary rigidity for Randers metrics


 If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M \times \partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem. We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points. The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric $g$, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow weights. This is handled by refining and extending our earlier results in the solenoidal gauge.

On the mean first arrival time of Brownian particles on Riemannian manifolds

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18].

Travel Time Tomography in Stationary Spacetimes

In this paper, we consider the boundary rigidity problem on a cylindrical domain in \({\mathbb {R}}^{1+n}\), \(n\ge 2\), equipped with a stationary (time-invariant) Lorentzian metric. We show that the time separation function between pairs of points on the boundary of the cylindrical domain determines the stationary spacetime, up to some time-invariant diffeomorphism, assuming that the metric satisfies some a-priori conditions.

Boundary rigidity of convex cocompact real hyperbolic manifolds

The goal of this short paper is to show that the hyperbolic metric is uniquely determined by the induced metric on the strictly convex smooth boundary for dimension ≥ 4 \geq 4 .

The boundary rigidity for holomorphic self-maps of some fibered domains

The boundary rigidity for holomorphic self-maps of some fibered domains

Axial force identification of space grid structural members using particle swarm optimization method

Space grid structures are popularly used in large-span civil structures. Identification of the axial internal forces and boundary conditions of the space grid structural members is very important. However, in the engineering practice, the boundary conditions of such members are not ideally rigid or pinned. They are sometimes semi-rigid conditions. Besides, direct measurements of the axial forces and the boundary rigidities of in-situ space grid structural members are difficult to make. Therefore, in this study, a Particle Swarm Optimization (PSO) algorithm-based axial force and boundary rigidity identification method is proposed for space grid structural members using multi-order natural frequencies, which can be easily obtained from in-situ tests. The theoretical background of the proposed method is discussed in detail, including the coupling relationship between the axial force and natural frequency and the PSO algorithm. The applicability and accuracy of the proposed method are validated through experimental tests and comparative analyses.