Boundary rigidity of Riemannian manifolds Plamen Stefanov and - - PDF document

Boundary rigidity

• f Riemannian manifolds

Plamen Stefanov and Gunther Uhlmann

Domain Ω ⊂ Rn, ∂Ω ∈ C∞. Let g = {gij} be a Riemannian metric in Ω. Distance function: ρg(x, y). Boundary rigidity: Does ρg(x, y), known for all x, y on ∂Ω, determine g, up to an isometry? In other words, if ρg1 = ρg2 on ∂Ω2, is there a diffeo ψ : Ω → Ω, ψ|∂Ω = Id, such that ψ∗g1 = g2? No, in general, but may be yes for simple met-

• rics. A metric g is simple in Ω, if the latter is

strictly convex w.r.t. g, and for any x ∈ ¯ Ω, the exp map is a diffeo on exp−1

x (¯

Ω).

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Equivalent formulation for (simple metrics): Know- ing the scattering relation σ, can we recover the metric g? σ : (x, ξ) → (y, η) This information is contained in the (hyperbo- lic) Dirichlet to Neumann map; in the scatter- ing kernel. Possible applications: in medical imaging, in geophysics, etc.

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Some history: Mukhometov; Mukhometov & Romanov, Bernstein & Gerver, Croke, Gromov, Michel, Pestov & Sharafutdinov Results for g conformal; flat; of negative cur- vature. S-Uhlmann ’98: for g close to the Euclidean

• ne.

Croke, Dairbekov and Sharafutdinov ’00: locally, near metrics with small enough curva- ture. Lassas, Sharafutdinov & Uhlmann ’03: one metric with small curvature, one close to the Euclidean. Pestov & Uhlmann ’03: n = 2, simple met- rics (no smallness assumptions)

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Linearized problem: Recover a tensor fij from the geodesic X-ray transform Igf(γ) =

• fij(γ(t))˙

γi(t)˙ γj(t) dt known for all max geodesics γ in Ω. Every tensor admits an orthogonal decompo- sition into a solenoidal part fs and a potential part dsv, f = fs + dsv, v|∂Ω = 0. Here δsfs = 0. The divergence δ is given by: [δf]i = gjk∇kfij. We have Ig(dsv) = 0. More precise formulation of the linearized problem: Does Igf = 0 imply fs = 0? We will call this s-injectivity of Ig. True at least for g Euclidean. Estimates?

• V. Sharafutdinov: if the curva-

ture is small enough, then Ig is s-injective and fs2

L2(Ω)

≤ C

• jνf|∂ΩH1/2(∂Ω) IgfL2(Γ−)

+ Igf2

H1(Γ−)

• .

Here jνf = fijνj, and ν is the normal.

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The small curvature condition was the largest known class of (simple) metrics with s-injective Ig. In case of 1-tensors (differential forms or vector fields) and functions, s-injectivity/injectivity is known for all simple metrics. Non-sharp stabil- ity estimates are also known and our methods allow us to obtain sharp estimates.

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A typical plan of attack is as follows: (1) injectivity of the linear problem (LP) (2) Stability (estimate) of the LP (3) Local uniqueness of the non-linear problem (NLP) (4) Stability estimate for the NLP (1) + (2) = ⇒ (3) + (4) In our case, injectivity of LP is s-injectivity; local uniqueness of NLP is mod isometry; and we show that (1) = ⇒ (2) + (3) + (4)

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Sketch of the main results (I):

• Study the linear problem in detail, show

that Ng := I∗

gIg is a ΨDO near Ω

• Find the principal symbol of Ng, identify

the kernel. Then Ng is elliptic on (Ker Ng)⊥

• Construct a parametrix of Ng on (Ker Ng)⊥

to recover fs, i.e., fs = ANgf + Kf. Note: the projection f → fs is not a ΨDO

• Prove an estimate of the type

fs ≤ CNgf∗ + CsfH−s, ∀s > 0

• If Ig is s-injective, show that

fs ≤ CNgf∗

• Show that s-injectivity of Ig implies local

uniqueness for the non-linear boundary rigi- dity problem near g.

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To illustrate the approach, consider a model problem: On a compact manifold M without boundary, assume that A is an elliptic pseudo- differential operator (ΨDO), i.e., locally, (Af)(x) = 1 (2π)n

• e−ix·ξa(x, ξ) ˆ

f(ξ) dξ. If a(x, ξ) = 0 for |ξ| ≫ 0, then a and A are called

• elliptic. Then one can construct a parametrix

B such that BA = Id + K, where K is smoothing, i.e., it sends “every- thing” into smooth functions. We construct B by iterations: B = B1 + B2 + . . . , where B1 is a ΨDO with symbol b = 1/a, etc.

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Now, since BA = Id + K, the problem of invertibility of A is reduced to that of Id+K, where K is compact. It is known that if the latter is injective (i.e., if −1 is not an eigenvalue of K), then (Id + K)−1 is bounded! Therefore, injectivity of A implies existence of A−1 and the estimate A−1 ≤ C. If, in addition, A = A(g) depends continuously

• n a parameter g (in our case, this is the metric

g), then K = K(g) has the same property, and C can be chosen locally uniform for g close to g0, under the condition that A(g0) is injective.

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Sketch of the main results (II): About the linear problem:

• Show that Ng is s-injective for real analytic

simple metrics using analytic ΨDO calcu- lus.

• Show that the constant C in

fs ≤ CNgf∗ is locally uniform in g, provided that g is near a metric for which Ig is s-injective.

• As a result show that Ig is injective (with

a stability estimate) for an open dense set G of metrics.

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Sketch of the main results (III): About the non-linear problem:

• Strong local uniqueness near g ∈ G
• H¨
• lder type of stability estimate near any

g ∈ G

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Representation for Ng: (Ngf)kl(x) = 1 √det g

• fij(y)

ρ(x, y)n−1 ∂ρ ∂yi ∂ρ ∂yj ∂ρ ∂xk ∂ρ ∂xl × det ∂2(ρ2/2) ∂x∂y dy, Here ρ(x, y) is the distance in the metric. Principal symbol of Ng: σp(Ng)ijkl(x, ξ) = cn |ξ|−1σ(εijεkl), where εij = δij −ξiξj/|ξ|2, and σ is symmetriza- tion, i.e., the average over all permutations of i, j, k, l. σp(Ng) is not elliptic, it vanishes on the range

• f σp(ds).

However, σp(Ng) is elliptic on the range of σp(δs).

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Define ∆s = δsds. Then v = vΩ = (∆s

D)−1δsf,

and fs = fs

Ω = f − ds(∆s D)−1δsf.

The Euclidean Case: Let g = e. Define vRn = (∆s)−1δsf, and fs

Rn = f − ds(∆s)−1δsf.

Assume Ief = 0. Then Nef = 0. ∃ parametrix A = A(D), such that ANef = fs

Rn =

⇒ fs

Rn = 0.

For x ∈ Ω, 0 = f = fs

Rn + dsvRn, therefore,

dsvRn = 0 there. This easily implies vRn = 0

• utside Ω, so

vRn = vΩ (!) (provided Ief = 0). Then fs

Rn = fs

Ω = 0, and

the s-injectivity of Ie is proved.

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For general simple metrics: ∃ parametrix A = A(x, D), such that f = ˜ fs + ds˜ v with ˜ fs := ANgf, and, loosely speaking, ˜ fs and ds˜ v have all prop- erties needed modulo smoothing operators (Ψ∞), except that ˜ v does not vanish on ∂Ω! (Not even mod Ψ∞.) They are analogues of fs

Rn

and vRn. To make ˜ v vanish on ∂Ω, we subtract a corrective term dsw (replace ˜ v by ˜ v − w) with w such that ∆sw = 0 in Ω, w|∂Ω = ˜ v|∂Ω. To get ˜ v|∂Ω, we use the fact that ds˜ v = − ˜ fs = −ANgf outside Ω, so ˜ v|∂Ω can be expressed as certain integrals of ANgf along geodesics connecting outside points with points on ∂Ω. This gives fs = A′Ngf in Ω mod Ψ∞. with A′ of order 2 (not 1, unfortunately).

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Let Ω1 ⊃⊃ Ω. In boundary local coordinates, set f ˜

H2(Ω1)

= xn∂nfH1(Ω1) +

n−1

• j=1

∂jfH1(Ω1) + fH1(Ω1). Theorem 1 (S-Uhlmann, ’03, ’04) Let g be simple, extended as a simple metric in Ω1. (a) The following estimate holds for each sym- metric 2-tensor f in H1(Ω): fs

ΩL2(Ω) ≤ CNgf ˜ H2(Ω1)+CsfH−s(Ω1),

∀s. (b) Ker Ig ∩ SL2(Ω) is finite dimensional and included in C∞(¯ Ω). (c) Assume that Ig is s-injective in Ω, i.e., that Ker Ig ∩ SL2(Ω) = {0}. Then for any symmet- ric 2-tensor f in H1(Ω) we have fsL2(Ω) ≤ CNgf ˜

H2(Ω1).

C is locally uniform as a function of g. (’04)

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The boundary rigidity problem: Theorem 2 (S-Uhlmann, ’03, ’04) Let g0 be a simple metric in Ω. Assume that Ig0 is s-

• injective. Then there exists ε > 0 and k > 0,

such that if for ˜ gj, j = 1, 2 we have ˜ gj − g0Ck ≤ ε, and ρg1 = ρg2

• n ∂Ω2,

then there exists a diffeomorphism ψ : ¯ Ω → ¯ Ω with ψ|Ω = Id, such that g2 = ψ∗g1. Sketch of the proof. Choose first semi-geodesic coordinates in Ω1, such that for g, ˜ g: gin = gni = δin, i = 1, . . . , n. Goal: under the assumption that Ig is s-injective, if ρ˜

g = ρg on ∂Ω2, and ˜

g is close to g, show that ˜ g = g (no additional diffeo).

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Linearize: ρ˜

g(x, y) − ρg(x, y) = Igf(x, y) + Rg(f)(x, y)

∀(x, y) ∈ ∂Ω2, where f = ˜ g − g is of the form fin = fni = 0. The remainder Rg is quadratic: |Rg(f)(x, y)| ≤ C|x−y|f2

C1(¯ Ω),

∀(x, y) ∈ ∂Ω2. If ρ˜

g = ρg, we get

|Igf(x, y)| ≤ C|x − y|f2

C1(¯ Ω),

∀(x, y) ∈ ∂Ω2. For f of the special form above, f ≤ CfsH2. Now, this, the estimate on Ngf from below in Thm 1, and interpolation inequalities imply f = 0 for f ≪ 1.

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Byproducts: X-ray transform of functions. Igf(γ) =

• f(γ(t)) dt

Mukhometov; Romanov; Bernstein and Gerver: Ig is injective for simple metrics. A non-sharp estimate is known. As a consequence of the injectivity: Theorem 3 Let g be a simple metric in Ω and assume that g is extended smoothly as a sim- ple metric near the convex domain Ω1 ⊃⊃ Ω. Then for any function f ∈ L2(Ω), f/C ≤ NgfH1(Ω1) ≤ Cf. Moreover, in Ω, f = c−1

n |D|χNgf mod H1(Ω).

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X-ray transform of 1-forms. Igf(γ) =

• fj(γ(t))˙

γj(t) dt As before, for each f = fjdxj, f = fs + dφ, δfs = 0, φ|∂Ω = 0. For simple metrics, Igf = 0 = ⇒ fs = 0 with a non-sharp estimate (Anikonov-Romanov). Theorem 4 Assume that g is simple metric in Ω and extend g as a simple metric in Ω1 ⊃⊃ Ω. Then for any 1-form f = fidxi in L2(Ω) we have fsL2(Ω) /C ≤ NgfH1(Ω1) ≤ C fsL2(Ω) . Moreover, fs = c−1

n |D|χNgf mod H1(Ω).

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Results for generic simple metrics

Results for the linear problem There was a large class of simple metrics (all with not so small curvature) for which s-injec- tivity of Ig and local (and global, of course) uniqueness of the rigidity problem were not known. We show next that this is true for an open dense set of such metrics. This is based on the following observations: (1) We have s-injectivity for any simple ana- lytic metric. (2) The metrics for which Ig is s-injective, form an open set. (3) Analytic metric are dense (of course) in the space of simple Ck metrics.

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The idea for proving (1) is to use analytic ΨDOs. Consider a model problem first: Let f be a function, not 2-tensor, and assume that Igf(γ) =

• f(γ(t)) dt = 0

∀γ. Assume that g is simple and analytic. Analytic weight function is also allowed. Then Ng is an analytic ΨDO (roughly speaking, a ΨDO with a real-analytic amplitude). Then one can construct a parametrix A to Ng = I∗

gIg, such

that ANgf = f + Rf in Ω1 ⊃⊃ Ω. with R analytic-regularizing, i.e., Rf is analytic ∀f. Assume now that Igf = 0. Then Ngf = 0, so f = −Rf in Ω1. The l.h.s. is compactly supported, the r.h.s. is

• analytic. Therefore, f = 0.

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Not such an impressive result, since we know that Ig is injective for functions, for all simple metrics by energy estimates (Mukhometov et al.) In case of analytic weight however, this is the way to prove injectivity and support theorems (Quinto and Boman ’87, ’91, Euclidean g, an- alytic weight). In our case things are more complicated be- cause we have non-Euclidean (analytic) met- ric, and we work with tensors. Injectivity is replaced by s-injectivity. We need recovery to infinite order at the boundary first. As a byproduct, one can generalize all of the in- tegral geometry results above to analytic sim- ple metrics and analytic weights.

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Formal formulation of the results above (for the linear problem) Theorem 5 (S-Uhlmann ’04) Let g be a sim- ple metric in Ω, real analytic in ¯ Ω. Then Ig is s-injective. Theorem 6 (S-Uhlmann ’04) ∃k0 such that for each k ≥ k0, the set Gk(Ω) of simple Ck(Ω) metrics in Ω for which Ig is s-injective is open and dense in the Ck(Ω) topology. Moreover, for any g ∈ Gk, fsL2(Ω) ≤ CNgf ˜

H2(Ω1),

∀f ∈ H1(Ω), with a constant C > 0 that can be chosen locally uniform in Gk in the Ck(Ω) topology.

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Generic results for the non-linear problem The s-injectivity of Ng and the stability estimate imply local uniqueness and H¨

• lder stability for

the non-linear problem. Generic local uniqueness: Theorem 7 (S-Uhlmann ’04) Let k0 and Gk(Ω) be as in Theorem 6. Then ∃k ≥ k0, such that ∀g0 ∈ Gk, ∃ε > 0, such that for any two metrics g1, g2 with gm − g0Ck(Ω) ≤ ε, m = 1, 2, we have the following: ρg1 = ρg2

• n (∂Ω)2

implies g2 = ψ∗g1 with some diffeomorphism ψ : Ω → Ω fixing the boundary.

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Stability estimate: Theorem 8 (S-Uhlmann ’04) Let k0 and Gk(M) be as in Theorem 6. Then for any µ < 1, there exits k ≥ k0 such that for any g0 ∈ Gk, there is an ε0 > 0 and C > 0 with the property that that for any two metrics g1, g2 with gm − g0C(Ω) ≤ ε0, and gmCk(M) ≤ A, m = 1, 2, with some A > 0, we have the fol- lowing stability estimate g2 − ψ∗g1C(Ω) ≤ C(A)ρg1 − ρg2µ

C(∂Ω×∂Ω)

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

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