On the boundary rigidity problem for surfaces Marco Mazzucchelli, - PowerPoint PPT Presentation

On the boundary rigidity problem for surfaces Marco Mazzucchelli, CNRS and ENS de Lyon (joint work with Colin Guillarmou and Leo Tzou) June 4, 2018

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM .

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y Boundary data:

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y Boundary data: ◮ Boundary distance d g := D g | ∂ M × ∂ M

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y Boundary data: ◮ Boundary distance d g := D g | ∂ M × ∂ M ◮ Lens data ( σ g , τ g )

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y Boundary data: ◮ Boundary distance d g := D g | ∂ M × ∂ M ◮ Lens data ( σ g , τ g ) τ g : ∂ in SM → [0 , ∞ ] τ g ( x , v ) = length of the geodesic γ v

Boundary data on compact Riemannian manifolds ( M , g ) compact Riemannian manifold, ∂ M � = ∅ φ t geodesic flow on unit tangent bundle SM . D g : M × M → [0 , ∞ ), D g ( x , y ) = g -distance from x to y Boundary data: ◮ Boundary distance d g := D g | ∂ M × ∂ M ◮ Lens data ( σ g , τ g ) τ g : ∂ in SM → [0 , ∞ ] τ g ( x , v ) = length of the geodesic γ v σ g : U ⊆ ∂ in SM → ∂ out SM σ g ( x , v ) = φ τ g ( x , v ) ( x , v )

Rigidity Question (boundary rigidity): does the boundary distance d g determine g ?

Rigidity Question (boundary rigidity): does the boundary distance d g determine g ? i.e. if d g 1 = d g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ?

Rigidity Question (boundary rigidity): does the boundary distance d g determine g ? i.e. if d g 1 = d g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ? Answer: No!

Rigidity Question (boundary rigidity): does the boundary distance d g determine g ? i.e. if d g 1 = d g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ? Answer: No! ( M , g ) invisible by d g

Rigidity Question (lens rigidity): do the lens data ( σ g , τ g ) determine g ?

Rigidity Question (lens rigidity): do the lens data ( σ g , τ g ) determine g ? i.e. if g 1 | ∂ M = g 2 | ∂ M , σ g 1 = σ g 2 , τ g 1 = τ g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ?

Rigidity Question (lens rigidity): do the lens data ( σ g , τ g ) determine g ? i.e. if g 1 | ∂ M = g 2 | ∂ M , σ g 1 = σ g 2 , τ g 1 = τ g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ? Answer: No!

Rigidity Question (lens rigidity): do the lens data ( σ g , τ g ) determine g ? i.e. if g 1 | ∂ M = g 2 | ∂ M , σ g 1 = σ g 2 , τ g 1 = τ g 2 , does there exists φ ∈ Diff ( M ) such that φ | ∂ M = id and φ ∗ g 2 = g 1 ? Answer: No! ( M , g s ) 1 − s s Lens data of ( M , g s ) independent of s ∈ [0 , 1]

Simple manifolds Michel’s conjecture (1981): Boundary rigidity holds on simple Riemannian manifolds (i.e. convex balls ( B n , g ) without conjugate points).

Simple manifolds Michel’s conjecture (1981): Boundary rigidity holds on simple Riemannian manifolds (i.e. convex balls ( B n , g ) without conjugate points). ◮ Croke-Otal, 1990: True if dim( M ) = 2 and g has negative curvature. ◮ Pestov-Uhlmann, 2004: True if dim( M ) = 2. ◮ Stefanov-Vasy-Uhlmann, 2017: True if g has negative sectional curvature.

Simple manifolds Michel’s conjecture (1981): Boundary rigidity holds on simple Riemannian manifolds (i.e. convex balls ( B n , g ) without conjugate points). ◮ Croke-Otal, 1990: True if dim( M ) = 2 and g has negative curvature. ◮ Pestov-Uhlmann, 2004: True if dim( M ) = 2. ◮ Stefanov-Vasy-Uhlmann, 2017: True if g has negative sectional curvature. Remark. On simple manifolds ( B n , g ), the scattering map σ g and the boundary distance d g are equivalent.

Rigidity on non-simple manifolds

Rigidity on non-simple manifolds ◮ Croke-Herreros, 2014: Lens rigidity holds for flat cylinders, flat M¨ obius strips, and negatively curved cylinders with convex boundary

Rigidity on non-simple manifolds ◮ Croke-Herreros, 2014: Lens rigidity holds for flat cylinders, flat M¨ obius strips, and negatively curved cylinders with convex boundary ◮ Guillarmou, 2015: If ( M 2 , g ) compact, convex, K g < 0, then � � � � ρ | ∂ M ≡ 0 e ρ g σ g determines M and the conformal class

Rigidity on non-simple manifolds ◮ Croke-Herreros, 2014: Lens rigidity holds for flat cylinders, flat M¨ obius strips, and negatively curved cylinders with convex boundary ◮ Guillarmou, 2015: If ( M 2 , g ) compact, convex, K g < 0, then � � � � ρ | ∂ M ≡ 0 e ρ g σ g determines M and the conformal class ◮ Burago-Ivanov, 2010: Boundary rigidity holds for nearly flat subdomains of R n .

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 .

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof (ingredients).

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof (ingredients). ◮ ( M , g ) as above, X-ray transform: � τ g ( x , v ) I : C 0 ( SM ) → L 1 ( ∂ in ) , If ( x , v ) = f ◦ φ t ( x , v ) dt . 0

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof (ingredients). ◮ ( M , g ) as above, X-ray transform: � τ g ( x , v ) I : C 0 ( SM ) → L 1 ( ∂ in ) , If ( x , v ) = f ◦ φ t ( x , v ) dt . 0 ◮ I m restriction of I to symmetric m tensors f ( x , v ) = F x ( v , ..., v ) � �� � × m

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof (ingredients). ◮ ( M , g ) as above, X-ray transform: � τ g ( x , v ) I : C 0 ( SM ) → L 1 ( ∂ in ) , If ( x , v ) = f ◦ φ t ( x , v ) dt . 0 ◮ I m restriction of I to symmetric m tensors f ( x , v ) = F x ( v , ..., v ) � �� � × m ◮ I 0 injective, I ∗ 0 surjective, ker I 1 = { exact 1-forms }

Rigidity on non-simple manifolds Theorem (Guillarmou, M., Tzou, 2017) Let ( M i , g i ) , i = 1 , 2 , compact oriented surfaces with no conjugate points, no trapped set, isometric boundaries, and same scattering map σ g 1 = σ g 2 . Then ∃ φ : M 1 → M 2 and ρ ∈ C ∞ ( M 1 ) such that ρ | ∂ M 1 ≡ 0 and φ ∗ g 2 = e ρ g 1 . Proof (ingredients). ◮ ( M , g ) as above, X-ray transform: � τ g ( x , v ) I : C 0 ( SM ) → L 1 ( ∂ in ) , If ( x , v ) = f ◦ φ t ( x , v ) dt . 0 ◮ I m restriction of I to symmetric m tensors f ( x , v ) = F x ( v , ..., v ) � �� � × m ◮ I 0 injective, I ∗ 0 surjective, ker I 1 = { exact 1-forms } � � � � h : M → C g -holomorphic ◮ σ g determines H g := h | ∂ M