Boundary rigidity of Riemannian manifolds Plamen Stefanov and - PDF document

Boundary rigidity of Riemannian manifolds Plamen Stefanov and Gunther Uhlmann

Domain Ω ⊂ R n , ∂ Ω ∈ C ∞ . Let g = { g ij } be a Riemannian metric in Ω. Distance function: ρ g ( x, y ). Boundary rigidity: Does ρ g ( x, y ) , known for all x , y on ∂ Ω , determine g , up to an isometry? In other words, if ρ g 1 = ρ g 2 on ∂ Ω 2 , is there a diffeo ψ : Ω → Ω, ψ | ∂ Ω = Id , such that ψ ∗ g 1 = g 2 ? 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). 1

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. 2

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 one. 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) 3

Linearized problem: Recover a tensor f ij from the geodesic X-ray transform � γ i ( t )˙ γ j ( t ) dt I g f ( γ ) = f ij ( γ ( t ))˙ known for all max geodesics γ in Ω . Every tensor admits an orthogonal decompo- sition into a solenoidal part f s and a potential part d s v , f = f s + d s v, v | ∂ Ω = 0 . Here δ s f s = 0. The divergence δ is given by: [ δf ] i = g jk ∇ k f ij . We have I g ( d s v ) = 0. More precise formulation of the linearized problem: Does I g f = 0 imply f s = 0? We will call this s-injectivity of I g . True at least for g Euclidean. Estimates? V. Sharafutdinov : if the curva- ture is small enough, then I g is s-injective and � f s � 2 � ≤ C � j ν f | ∂ Ω � H 1 / 2 ( ∂ Ω) � I g f � L 2 (Γ − ) L 2 (Ω) + � I g f � 2 � . H 1 (Γ − ) Here j ν f = f ij ν j , and ν is the normal. 4

The small curvature condition was the largest known class of (simple) metrics with s-injective I g . 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. 5

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) 6

Sketch of the main results (I): • Study the linear problem in detail, show that N g := I ∗ g I g is a ΨDO near Ω • Find the principal symbol of N g , identify the kernel. Then N g is elliptic on (Ker N g ) ⊥ • Construct a parametrix of N g on (Ker N g ) ⊥ to recover f s , i.e., f s = AN g f + Kf . Note: the projection f �→ f s is not a ΨDO • Prove an estimate of the type � f s � ≤ C � N g f � ∗ + C s � f � H − s , ∀ s > 0 • If I g is s-injective, show that � f s � ≤ C � N g f � ∗ • Show that s-injectivity of I g implies local uniqueness for the non-linear boundary rigi- dity problem near g . 7

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, 1 � e − ix · ξ a ( x, ξ ) ˆ ( Af )( x ) = f ( ξ ) dξ. (2 π ) n 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 = B 1 + B 2 + . . . , where B 1 is a ΨDO with symbol b = 1 /a , etc. 8

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 on 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 g 0 , under the condition that A ( g 0 ) is injective. 9

Sketch of the main results (II): About the linear problem: • Show that N g is s-injective for real analytic simple metrics using analytic ΨDO calcu- lus. • Show that the constant C in � f s � ≤ C � N g f � ∗ is locally uniform in g , provided that g is near a metric for which I g is s-injective. • As a result show that I g is injective (with a stability estimate) for an open dense set G of metrics. 10

Sketch of the main results (III): About the non-linear problem: • Strong local uniqueness near g ∈ G • H¨ older type of stability estimate near any g ∈ G 11

Representation for N g : f ij ( y ) 1 ∂ρ ∂ρ ∂ρ ∂ρ � ( N g f ) kl ( x ) = √ det g ρ ( x, y ) n − 1 ∂y i ∂y j ∂x k ∂x l × det ∂ 2 ( ρ 2 / 2) dy, ∂x∂y Here ρ ( x, y ) is the distance in the metric. Principal symbol of N g : σ p ( N g ) ijkl ( x, ξ ) = c n | ξ | − 1 σ ( ε ij ε kl ) , where ε ij = δ ij − ξ i ξ j / | ξ | 2 , and σ is symmetriza- tion, i.e., the average over all permutations of i, j, k, l . σ p ( N g ) is not elliptic, it vanishes on the range of σ p ( d s ). However, σ p ( N g ) is elliptic on the range of σ p ( δ s ). 12

Define ∆ s = δ s d s . Then D ) − 1 δ s f, v = v Ω = (∆ s and f s = f s D ) − 1 δ s f. Ω = f − d s (∆ s The Euclidean Case: Let g = e . Define v R n = (∆ s ) − 1 δ s f, and R n = f − d s (∆ s ) − 1 δ s f. f s Assume I e f = 0. Then N e f = 0. ∃ parametrix A = A ( D ), such that AN e f = f s ⇒ f s R n = R n = 0. For x �∈ Ω, 0 = f = f s R n + d s v R n , therefore, d s v R n = 0 there. This easily implies v R n = 0 outside Ω, so v R n = v Ω (!) (provided I e f = 0). Then f s R n = f s Ω = 0, and the s-injectivity of I e is proved. 13

For general simple metrics: ∃ parametrix A = A ( x, D ), such that f s + d s ˜ f s := AN g f, f = ˜ ˜ v with f s and d s ˜ and, loosely speaking, ˜ v have all prop- erties needed modulo smoothing operators (Ψ ∞ ), except that ˜ v does not vanish on ∂ Ω! (Not They are analogues of f s even mod Ψ ∞ .) R n and v R n . To make ˜ v vanish on ∂ Ω, we subtract a corrective term d s w (replace ˜ v by ˜ v − w ) with w such that ∆ s w = 0 in Ω , w | ∂ Ω = ˜ v | ∂ Ω . f s = v | ∂ Ω , we use the fact that d s ˜ v = − ˜ To get ˜ − AN g f outside Ω, so ˜ v | ∂ Ω can be expressed as certain integrals of AN g f along geodesics connecting outside points with points on ∂ Ω. This gives f s = A ′ N g f in Ω mod Ψ ∞ . with A ′ of order 2 (not 1, unfortunately). 14

Let Ω 1 ⊃⊃ Ω. In boundary local coordinates, set n − 1 = � x n ∂ n f � H 1 (Ω 1 ) + � � f � ˜ � ∂ j f � H 1 (Ω 1 ) H 2 (Ω 1 ) j =1 + � f � H 1 (Ω 1 ) . Theorem 1 (S-Uhlmann, ’03, ’04) Let g be simple, extended as a simple metric in Ω 1 . (a) The following estimate holds for each sym- metric 2-tensor f in H 1 (Ω) : � f s Ω � L 2 (Ω) ≤ C � N g f � ˜ H 2 (Ω 1 ) + C s � f � H − s (Ω 1 ) , ∀ s. (b) Ker I g ∩ S L 2 (Ω) is finite dimensional and included in C ∞ (¯ Ω) . (c) Assume that I g is s-injective in Ω , i.e., that Ker I g ∩ S L 2 (Ω) = { 0 } . Then for any symmet- ric 2-tensor f in H 1 (Ω) we have � f s � L 2 (Ω) ≤ C � N g f � ˜ H 2 (Ω 1 ) . C is locally uniform as a function of g . (’04) 15

The boundary rigidity problem: Theorem 2 (S-Uhlmann, ’03, ’04) Let g 0 be a simple metric in Ω . Assume that I g 0 is s- injective. Then there exists ε > 0 and k > 0 , such that if for ˜ g j , j = 1 , 2 we have � ˜ g j − g 0 � C k ≤ ε, and on ∂ Ω 2 , ρ g 1 = ρ g 2 then there exists a diffeomorphism ψ : ¯ Ω → ¯ Ω with ψ | Ω = Id, such that g 2 = ψ ∗ g 1 . Sketch of the proof. Choose first semi-geodesic coordinates in Ω 1 , such that for g , ˜ g : g in = g ni = δ in , i = 1 , . . . , n. Goal: under the assumption that I g is s-injective, g = ρ g on ∂ Ω 2 , and ˜ if ρ ˜ g is close to g , show that ˜ g = g (no additional diffeo). 16

Linearize: ρ ˜ g ( x, y ) − ρ g ( x, y ) = I g f ( x, y ) + R g ( f )( x, y ) ∀ ( x, y ) ∈ ∂ Ω 2 , where f = ˜ g − g is of the form f in = f ni = 0. The remainder R g is quadratic: | R g ( f )( x, y ) | ≤ C | x − y |� f � 2 ∀ ( x, y ) ∈ ∂ Ω 2 . Ω) , C 1 (¯ If ρ ˜ g = ρ g , we get | I g f ( x, y ) | ≤ C | x − y |� f � 2 ∀ ( x, y ) ∈ ∂ Ω 2 . Ω) , C 1 (¯ For f of the special form above, � f � ≤ C � f s � H 2 . Now, this, the estimate on � N g f � from below in Thm 1, and interpolation inequalities imply f = 0 for � f � ≪ 1. 17