Boundary Rigidity Research Articles

Recovery of the C ∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the “boundary rigidity problem”. If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the Taylor series of the metric tensor (C ∞ jet) at the boundary (see Lassas et al. (Math Ann 325:767–793, 2003), Uhlmann et al. (Adv Appl Math 31:379–387, 2003)). In this paper we give a new reconstruction procedure for the C ∞ jet at non-concave points on the boundary using the localized boundary distance function. A closely related problem is the “lens rigidity problem”, which asks whether the lens data determine metric structure uniquely. Lens data include information of boundary distance function, the lengths of all geodesics, and the locations and directions where geodesics hit the boundary. We give the first examples that show that lens data cannot uniquely determine the C ∞ jet. The example include two manifolds with the same boundary and the same lens data, but different C ∞ jets. With some additional careful work, we can find examples with different C 1 jets, which means the boundaries in the two lens-equivalent manifolds have different second fundamental forms.

On Gromov–Hausdorff stability in a boundary rigidity problem

Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance function of $M$ is $C^0$-close to that of $D$, the volumes of $M$ and $D$ are almost equal, and volumes of metric balls in $M$ have a certain lower bound in terms of radius.

Boundary rigidity and filling volume minimality of metrics close to a flat one

We say that a Riemannian manifold (M, g) with a non-empty boundary ∂M is a minimal orientable filling if, for every compact orientable (M, g) with ∂M = ∂M, the inequality d g (x, y) ≥ d g (x, y) for all x, y ∈ ∂M implies vol(M, g) ≥ vol(M, g). We show that if a metric g on a region M C R n with a connected boundary is sufficiently C 2 -close to a Euclidean one, then it is a minimal filling. By studying the equality case vol(M, g) = vol(M, g) we show that if d g (x, y) = d g (x, y) for all x, y ∈ ∂M then (M, g) is isometric to (M, g). This gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of Michel's conjecture.

Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.

Boundary rigidity for some classes of meromorphic functions

Let f be a function meromorphic on the open unit disk D , with angular boundary limits bounded by one in modulus almost everywhere on the unit circle. We give sufficient conditions in terms of boundary asymptotics at finitely many points on the unit circle T for f to be a ratio of two finite Blaschke products. A necessary condition is that f has finitely many poles in D , i.e., that f is a generalized Schur function. Similar rigidity statements are presented for generalized Caratheodory and generalized Nevanlinna functions. Mathematics subject classification (2000): 30E05.

Some smooth Finsler deformations of hyperbolic surfaces

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.

A new proof of the rigidity problem

In this short note we give a new proof of the boundary rigidity problem in a Euclidean setting proved by Croke. Our method is based on the differentiability of Busemann functions and the characteristic of Euclidean metric on Riemannian manifolds without conjugate points.

The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α. For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α. We also show injectivity and stability for g and α in a generic class G including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple ( g , α ) , rigidity for metrics in a given conformal class, and locally, near any ( g , α ) ∈ G . We also show that simple magnetic systems on two-dimensional manifolds are always rigid.

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λg: C∞ (∂M) → C∞(∂M) on a compact Riemannian manifold (M, g) with boundary is defined by Λgh = ∂u/∂v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦∂M = h and v is the unit normal to the boundary. If gt = g + tƒ is a variation of the metric g by a symmetric tensor field ƒ, then Λg t = Λg + tΛƒ + o(t). We study the question: How do tensor fields ƒ look like for which Λƒ =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.

Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.

Boundary rigidity and stability for generic simple metrics

We study the boundary rigidity problem for compact Riemannian manifolds with boundary ( M , g ) (M,g) : is the Riemannian metric g g uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ g ( x , y ) \rho _g(x,y) known for all boundary points x x and y y ? We prove in this paper local and global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set G \mathcal {G} of simple Riemannian metrics such that for any g 0 ∈ G g_0\in \mathcal {G} , any two Riemannian metrics in some neighborhood of g 0 g_0 having the same distance function, must be isometric. Similarly, there is a generic set of pairs of simple metrics with the same property. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in G \mathcal {G} .

Recent progress on the boundary rigidity problem

The boundary rigidity problem consists in determining a compact, Riemannian manifold with boundary, up to isometry, by knowing the boundary distance function between boundary points. In this paper we announce the result of our forthcoming article that one can solve this problem for generic simple metrics. Moreover we probe stability estimates for this problem.

Transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic saturated soil

Based on Biot's wave equation, this paper discusses the transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic saturated soil. The analytical solution is derived for the transient response to an axisymmetric surface load and fluid pressure in Laplace transform domain. Numerical results are obtained by inverting the Laplace transform presented by Durbin, and are used to analyze the influences of the partial permeable property of boundary and relative rigidity of shell and soil on the transient response of the spherical cavity. It is shown that the influence of these two parameters is remarkable. The available solutions of permeable and impermeable boundary without shell are only two extreme cases of this paper.

Stability estimates for the X-ray transform of tensor fields and boundary rigidity

We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

BOUNDARY RIGIDITY AND HOLOGRAPHY

We review boundary rigidity theorems assessing that, under appropriate conditions, Riemannian manifolds with the same spectrum of boundary geodesics are isometric. We show how to apply these theorems to the problem of reconstructing a $d+1$ dimensional, negative curvature space-time from boundary data associated to two-point functions of high-dimension local operators in a conformal field theory. We also show simple, physically relevant examples of negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity theorems. In those examples, we explicitly show that the spectrum of boundary geodesics is not sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality.

Semiglobal boundary rigidity for Riemannian metrics

Let (M, g) be a Riemannian manifold with boundary. In this paper we consider the problem of determining the metric g from its associated boundary distance function dg(x, y) = dist (x, y), x, y ∈ ∂M, that is the geodesic distance between boundary points. This problem arose in rigidity questions in Riemannian geometry [12], [6], [8]. For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one, this problem is called the inverse kinematic problem which arose in Geophysics and has a long history starting at least in the early part of the 20th century with Herglotz [10]. He considered the case where M is a ball {x ∈ R | r = |x| ≤ R} equipped with a spherically symmetric metric ds2 = dx2/c2(r) where c(r) is a positive function depending only on the radius r = |x|. Herglotz found a formula to determine c(r) from the boundary distance function. Physically this corresponds to the case of a spherically symmetric Earth model with an index of refraction depending only on the radius. The boundary distance function corresponds to the travel times of e.g. acoustic waves going through the Earth and measured at the surface. The general problem for the case that the sound speed depends on all variables has been extensively studied (see for instance [17] and the references given there). Also, this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements, see [20].

Uniqueness in Inverse Problems for Elastic Media with Residual Stress

We consider dynamic inverse problems for bounded, three-dimensional elastic media with residual stress. In fact, we consider more general (nonclassical) linear hyperelastic media with the property that wave propagation occurs along the geodesics of Riemannian metrics. We show that the travel times of wave propagation through the object, together with entry and exit locations and directions, are determined by the Dirichlet-to-Neumann map in the absence of caustics and when the characteristic cones do not intersect. It follows by a boundary rigidity result of Lassas et al. (Lassas, M., Sharafutdinov, V., Uhlmann, G. ([2003]). Semiglobal boundary rigidity for Riemannian metrics. Math. Ann. 325(4):767–793.) that, under certain conditions, the geometry of the wave paths through the medium is determined up to the pullback by diffeomorphisms that fix the boundary. We also prove that the Dirichlet-to-Neumann map determines the six independent parameters at the boundary that describe the material properties of elastic media with residual stress.

Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the non-removable intersection between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

Boundary Rigidity for Lagrangian Submanifolds, Non-Removable Intersections, and Aubry—Mather Theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the non-removable intersection between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

Boundary Permeability as a Cultural Dimension

Interviews conducted with American and Israeli managers and engineers who work together in the Silicon Valley, California, revealed that many of the cultural differences and difficulties encountered in cross cultural working relationships between members of the two cultures represent a single theme. The Americans were perceived and described as more compartmentalized and bounded in many respects, while the Israelis were perceived and described as more unbounded and as tending to disregard or cross boundaries. This theme repeated itself with regard to several types of boundaries: expressive boundaries, bureaucratic boundaries, temporal boundaries, and boundaries between work and non-work activities and relationships. These differences affected the quality of communication and the level of trust and cooperation between members of the two groups. Following these findings, and the observation that many of the cultural dimensions commonly used by social scientists can be conceived of in terms of differences in boundary permeability, we propose that cultures may differ on a meta-dimension of boundary rigidity versus permeability, and that such differences are consequential for cross cultural relations in organizations.