ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Pages 64–70 (June 23, 2005) S 1079-6762(05)00148-4

RECENT PROGRESS ON THE BOUNDARY RIGIDITY PROBLEM

PLAMEN STEFANOV AND GUNTHER UHLMANN

(Communicated by Dmitri Burago)

Abstract. 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.

1. Main results

Let (M, ∂M, g) be a compact Riemannian manifold with boundary. Denote by ρ_{g} the distance function in the metric g. We consider the inverse problem of whether ρ_{g}(x, y), known for all x, y on ∂M, determines the metric uniquely. It is clear that any isometry which is the identity at the boundary will give rise to the same distance functions on the boundary. Therefore, the natural question is whether this is the only obstruction to uniqueness. This is known in differential geometry as the boundary rigidity problem. The boundary distance function only takes into account the shortest paths, and it is easy to find counterexamples where ρ_{g} does not carry any information about certain open subset of M, so one needs to pose some restrictions on the metric. One such condition is simplicity of the metric.

Definition 1.1. We say that the Riemannian metric g is simple in M, if ∂M is strictly convex with respect to g, and for any x ∈ M, the exponential map exp_{x} : exp_{x}^{−1}(M) → M is a diffeomorphism.

Michel [13] conjectured that a simple metric g is uniquely determined, up to an action of a diffeomorphism fixing the boundary, by the boundary distance function ρ_{g}(x, y) known for all x and y on ∂M.

This problem also arose in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves. It goes back to Herglotz [11] and Wiechert and Zoeppritz [30]. Although the emphasis has been on the case that the medium is isotropic, the anisotropic case has been of interest in geophysics since it has been found that the inner core of the Earth exhibits anisotropic behavior [6].

\begin{matrix}c \\ ◯ \end{matrix}2005 American Mathematical Society

Reverts to public domain 28 years from publication

Received by the editors March 8, 2005.

2000 Mathematics Subject Classification. Primary 53C20.

Key words and phrases. Boundary rigidity, Riemannian manifold, inverse problem.

The first author was supported in part by NSF Grant DMS-0400869.

The second author was supported in part by NSF Grant DMS-0245414.

Note that a simple metric g in M can be extended to a simple metric in some M_{1} with M ⊂⊂ M_{1}. If we fix x = x_{0} ∈ M above, we also obtain that each simple manifold is diffeomorphic to a (strictly convex) domain Ω ⊂ R^{n} with the Euclidean coordinates x in a neighborhood of Ω and a metric g(x) there. For this reason, it is enough to prove our results for domains Ω in R^{n}.

A closely related problem is to recover g from the scattering relation that maps initial points x ∈ ∂M and directions ξ of maximal geodesics through M into outgoing points y ∈ ∂M and directions η. Weaker geometric assumptions are needed to formulate this problem, and in the case of simple metrics, it is equivalent to the boundary rigidity problem. The scattering relation describes propagation of singularities of the corresponding wave equation through M, and is encoded in the hyperbolic Dirichlet-to Neumann map, and in the scattering operator in the case when M is embedded in R^{n}.

Unique recovery of g (up to an action of a diffeomorphism) is known for simple metrics conformal to each other [8], [4], [14], [15], [16], [2], for flat metrics [10], for locally symmetric spaces of negative curvature [3]. In two dimensions it was known for simple metrics with negative curvature [7] and [17], and recently it was shown in [19] for simple metrics with no restrictions on the curvature. In [23], the authors proved a local result for metrics in a small neighborhood of the Euclidean one. This result was used in [12] to prove a semiglobal solvability result. Burago and Ivanov have shown recently that metrics close to the Euclidean metric are boundary rigid [5].

It is known [20], that a linearization of the boundary rigidity problem near a simple metric g is given by the following integral geometry problem: recover a symmetric tensor of order 2, which in any coordinates is given by f = (f_{ij}), by the geodesic X-ray transform

I_{g}f(γ) = ∫ f_{ij}(γ(t))γ̇^{i}(t)γ̇^{j}(t) dt

known for all geodesics γ in M. It can be easily seen that I_{g}dv = 0 for any vector field v with v|_{∂M} = 0, where dv denotes the symmetric differential

(1.1)  [dv]_{ij} = \frac{1}{2} (∇_{i}v_{j} + ∇_{j}v_{i}) ,

and ∇_{k}v denote the covariant derivatives of the vector field v. This is the linear version of the fact that ρ_{g} does not change on (∂M)^{2} := ∂M × ∂M under an action of a diffeomorphism as above. The natural formulation of the linearized problem is therefore that I_{g}f = 0 implies f = dv with v vanishing on the boundary. We will refer to this property as s-injectivity of I_{g}. More precisely, we have.

Definition 1.2. We say that I_{g} is s-injective in M, if I_{g}f = 0 and f ∈ L^{2}(M) imply f = dv with some vector field v ∈ H_{0}^{1}(M).

Any symmetric tensor f ∈ L^{2}(M) admits an orthogonal decomposition f = f^{s} + dv into a solenoidal and potential parts with v ∈ H_{0}^{1}(M), and f^{s} divergence free, i.e., δf^{s} = 0, where δ is the adjoint operator to −d given by [δf]_{i} = g^{jk}∇_{k}f_{ij}. Therefore, I_{g} is s-injective, if it is injective on the space of solenoidal tensors.

The inversion of I_{g} is a problem of independent interest in integral geometry, and our first two theorems are related to it. The s-injectivity of I_{g} was proved in [18] for metrics with negative curvature, in [20] for metrics with small curvature, and in [22] for Riemannian surfaces with no focal points. A conditional and non-sharp stability estimate for metrics with small curvature is also established in [20]. This estimate was used in [9] to get local uniqueness results for the boundary rigidity problem under the same condition. In [24], we proved stability estimates for sinjective metrics (see (1.2) below) and sharp estimates about the recovery of a 1-form f = f_{j}dx^{j} and a function f from the associated I_{g}f. The stability estimates proven in [24] were used to prove local uniqueness for the boundary rigidity problem near any simple metric g with s-injective I_{g}.

Similarly to [28], we say that f is analytic in the set K (not necessarily open), if it is real analytic in some neighborhood of K. Our first main result is about s-injectivity at simple analytic metrics.

Theorem 1.3. Let g be a simple, real analytic metric in M. Then I_{g} is s-injective.

As shown in [24], the s-injectivity of I_{g} for analytic simple g implies a stability estimate for I_{g}. In the next theorem we show something more, namely that we have a stability estimate for g in a neighborhood of each analytic metric, which leads to stability estimates for generic metrics.

Let M_{1} ⊃ M be a compact manifold which is a neighborhood of M, and suppose g extends as a simple metric there. We always assume that our tensors are extended as zero outside M, which may create jumps at ∂M. Define the normal operator N_{g} = I_{g}^{∗}I_{g}, where I_{g}^{∗} denotes the operator adjoint to I_{g} with respect to an appropriate measure. We showed in [24] that N_{g} is a pseudodifferential operator in M_{1} of order −1.

As in [24], we define the space H̃^{2}(M_{1}) that in particular satisfies H^{2}(M_{1}) ⊂ H̃^{2}(M_{1}) ⊂ H^{1}(M_{1}) (we refer to [24] for details). On the other hand, f ∈ H^{1}(M) implies N_{g}f ∈ H̃^{2}(M_{1}) despite the possible jump of f at ∂M.

Theorem 1.4. There exists k_{0} such that for each k ≥ k_{0}, the set G^{k}(M) of simple C^{k}(M) metrics in M for which I_{g} is s-injective is open and dense in the C^{k}(M) topology. Moreover, for any g ∈ G^{k},

(1.2)  ∥f^{s}∥_{L^{2}(M)} ≤ C∥N_{g}f∥_{H̃^{2}(M_{1})}, ∀f ∈ H^{1}(M),

with a constant C > 0 that can be chosen locally uniform in G^{k} in the C^{k}(M) topology.

Of course, G^{k} includes all real analytic simple metrics in M, according to Theorem 1.3.

The analysis of I_{g} can also be carried out for symmetric tensors of any order; see e.g. [20] and [21]. Since we are motivated by the boundary rigidity problem, and to simplify the exposition, we study only tensors of order 2.

Theorem 1.4 and especially estimate (1.2) allow us to prove the following local generic uniqueness result for the non-linear boundary rigidity problem.

Theorem 1.5. Let k_{0} and G^{k}(M) be as in Theorem 1.4. There exists k ≥ k_{0} such that for any g_{0} ∈ G^{k}, there is ε > 0 such that for any two metrics g_{1}, g_{2} with ∥g_{m} − g_{0}∥_{C^{k}(M)} ≤ ε, m = 1, 2, we have the following:

(1.3)  ρ_{g_{1}} = ρ_{g_{2}} on (∂M)^{2} implies g_{2} = ψ_{∗}g_{1}

with some C^{k+1}(M)-diffeomorphism ψ : M → M fixing the boundary.

Finally, we prove a conditional stability estimate of Hölder type. A similar estimate near the Euclidean metric was proven in [29] based on the approach in Theorem 1.6. Let k_{0} and G^{k}(M) be as in Theorem 1.4. Then for any μ < 1, there exists k ≥ k_{0} such that for any g_{0} ∈ G^{k}, there is an ε_{0} > 0 and C > 0 with the property that that for any two metrics g_{1}, g_{2} with ∥g_{m} − g_{0}∥_{C(M)} ≤ ε_{0}, and ∥g_{m}∥_{C^{k}(M)} ≤ A, m = 1, 2, with some A > 0, we have the following stability estimate:

∥g_{2} − ψ_{∗}g_{1}∥_{C^{2}(M)} ≤ C(A)∥ρ_{g_{1}} − ρ_{g_{2}}∥_{C(∂M×∂M)}^{μ}

with some diffeomorphism ψ : M → M fixing the boundary.

Theorem 1.6 can be used to obtain stability near generic simple metrics for the inverse problem of recovering g from the hyperbolic Dirichlet-to-Neumann map Λ_{g}. It is known that g can be recovered uniquely from Λ_{g}, up to a diffeomorphism as above, see e.g. [1]. This result however relies on a unique continuation theorem by Tataru [27] and it is unlikely to provide Hölder type of stability estimate as above. By using the fact that ρ_{g} is related to the leading singularities in the kernel of Λ_{g}, we can prove a Hölder stability estimate under the assumptions above, relating g and Λ_{g}. We refer to [25] for details.

2. Sketch of the main ideas

As mentioned above, we can assume that M is an open bounded domain Ω ⊂ R^{n} with smooth boundary, and g is a simple Riemannian metric there.

2.1. S-injectivity for analytic metrics, Theorem 1.3. The proof of Theorem 1.3 is based on the following. For analytic metrics, the normal operator N_{g} = I_{g}^{∗}I_{g} is an analytic pseudodifferential operator with a non-trivial null space. We construct an analytic parametrix that allows us to reconstruct the solenoidal part of a tensor field from its geodesic X-ray transform, up to a term that is analytic near Ω. If I_{g}f = 0, we show that for some v vanishing on ∂Ω, f̃ := f − dv must be flat at ∂Ω and analytic in Ω̄, hence f̃ = 0. This is similar to the known argument that an analytic elliptic pseudodifferential operator resolves the analytic singularities, hence cannot have compactly supported functions in its kernel. In our case we have a non-trivial kernel, and complications due to the presence of a boundary, in particular loss of one derivative.

2.2. The a priori linear stability estimate, Theorem 1.4. The proof of the basic estimate (1.2) is based on the following ideas. For g of finite smoothness, one can still construct a parametrix Q_{g} of N_{g} as above that allows us to reconstruct f^{s} from N_{g}f up to smoothing operator terms. This is done in a way similar to that in [24] in two steps: first we invert microlocally N_{g} in a neighborhood Ω_{1} of Ω, and that gives us f_{Ω_{1}}^{s}, i.e., the solenoidal projection of f but related to Ω_{1}. Next, we compare f_{Ω_{1}}^{s} and f^{s} and show that one can get the latter from the former by an operator that loses one derivative. This is the same construction as in 2.1 above but the metric is only C^{k}, k ≫ 1.

After applying the parametrix Q_{g}, the equation for recovering f^{s} from N_{g}f is reduced to solving the Fredholm equation

(2.1)  (S_{g} + K_{g})f = Q_{g}N_{g}f, f ∈ S_{g}L^{2}(Ω),

where S_{g} is the projection to solenoidal tensors; similarly we denote by P_{g} the projection onto potential tensors. Here K_{g} is a compact operator on S_{g}L^{2}(Ω). We can write this as an equation in the whole L^{2}(Ω) by adding P_{g}f to both sides above to get

(2.2)  (I + K_{g})f = (Q_{g}N_{g} + P_{g})f.

Then the solenoidal projection of the solution of (2.2) solves (2.1). A finite rank modification of K_{g} above can guarantee that I + K_{g} has a trivial kernel, and therefore is invertible, if and only if N_{g} is s-injective. The problem then reduces to that of invertibility of I + K_{g}. The operators above depend continuously on g ∈ C^{k}, k ≫ 1. Since for g analytic, I + K_{g} is invertible by Theorem 1.3, it would still be invertible in a neighborhood of any analytic g, and estimate (1.2) is true with a locally uniform constant. Analytic (simple) metrics are dense in the set of all simple metrics, and this completes the sketch of the proof of Theorem 1.4.

2.3. Generic local and global boundary rigidity, Theorem 1.5. We prove Theorem 1.5 by linearizing and using Theorem 1.4, and especially (1.2); see also [24]. This requires first to pass to special semigeodesic coordinates related to each metric in which g_{in} = δ_{in}, ∀i. We denote the corresponding pull-backs by g_{1}, g_{2} again. Then we show that if g_{1} and g_{2} have the same distance on the boundary, then g_{1} = g_{2} on the boundary with all derivatives. As a result, for f := g_{1} − g_{2} we get that f ∈ C_{0}^{l}(Ω̄) with l ≫ 1 if k ≫ 1; and f_{in} = 0, ∀i. Then we linearize to get

∥N_{g_{1}}f∥_{L^{∞}(Ω_{1})} ≤ C∥f∥_{C^{1}}^{2},

where Ω_{1} ⊃ Ω̄ is as above. Combine this with (1.2) and interpolation estimates, to get ∀μ < 1,

∥f^{s}∥_{L^{2}} ≤ C∥f∥_{L^{2}}^{1+μ}.

One can show that tensors satisfying f_{in} = 0 also satisfy ∥f∥_{L^{2}} ≤ C∥f^{s}∥_{H^{2}}, and using this, and interpolation again, we get

∥f∥_{L^{2}} ≤ C∥f∥_{L^{2}}^{1+μ′}, μ′ > 0.

This implies f = 0 for ∥f∥ ≪ 1. Note that the condition f ∈ C_{0}^{l}(Ω̄) is used to make sure that f, extended as zero in Ω_{1} \ Ω, is in H_{0}^{l}(Ω), and then use this fact in the interpolation estimates.

2.4. The stability estimate. To prove Theorem 1.6, we basically follow the uniqueness proof sketched above by showing that each step is stable. The analysis is more delicate near pairs of points too close to each other. An important ingredient of the proof is stability at the boundary, which is also of independent interest:

Theorem 2.1. Let g_{0} and g_{1} be two simple metrics in Ω, and Γ ⊂⊂ Γ′ ⊂ ∂Ω be two sufficiently small open subsets of the boundary. Then for some diffeomorphism ψ fixing the boundary,

∥∂_{x^{n}}^{k}(ψ_{∗}g_{1} − g_{0})∥_{C^{m}(Γ̄)} ≤ C_{k,m}∥ρ_{g_{1}}^{2} − ρ_{g_{0}}^{2} ∥_{C^{m+2k+2}(\overline{Γ′×Γ′})},

where C_{k,m} depends only on Ω and on an upper bound of g_{0}, g_{1} in C^{m+2k+5}(Ω̄).

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Department of Mathematics, Purdue University, West Lafayette, IN 47907

E-mail address: stefanov@math.purdue.edu

Department of Mathematics, University of Washington, Seattle, WA 98195

E-mail address: gunther@math.washington.edu