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
Recommend
More recommend