scholarly journals Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

10.53733/191 ◽  
2021 ◽  
Vol 52 ◽  
pp. 733-763
Author(s):  
Nikolas Eptaminitakis ◽  
Robin Graham
Keyword(s):  
Boundary Point ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
X Ray ◽  
Ray Transform ◽  
Asymptotically Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

scholarly journals X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds

2020 ◽  
Vol 69 (7) ◽  
pp. 2857-2919 ◽  
Author(s):  
C. Robin Graham ◽  
Colin Guillarmou ◽  
Plamen Stefanov ◽  
Gunther Uhlmann
Keyword(s):  
Hyperbolic Manifolds ◽  
X Ray ◽  
Ray Transform ◽  
Asymptotically Hyperbolic Manifolds ◽  
Boundary Rigidity ◽  
Asymptotically Hyperbolic

scholarly journals Sharp Resolvent Estimates on Non-positively Curved Asymptotically Hyperbolic Manifolds

2020 ◽  
Author(s):  
Yiran Wang
Keyword(s):  
Energy Estimate ◽  
Weighted Sobolev Spaces ◽  
High Energy ◽  
Oscillatory Behavior ◽  
Hyperbolic Manifolds ◽  
Resolvent Estimates ◽  
Positive Sectional Curvature ◽  
Asymptotically Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic ◽  
Positively Curved

Abstract We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHMs). In the literature, estimates of $R(\lambda )$ on weighted Sobolev spaces of the order $O(|\lambda |^{N})$ were established for some $N> -1$, $|\lambda |$ large, and $\lambda \in{{\mathbb{C}}}$ in strips where $R(\lambda )$ is holomorphic. In this work, we prove the optimal bound $O(|\lambda |^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.

scholarly journals Inverse scattering on asymptotically hyperbolic manifolds

2000 ◽  
Vol 184 (1) ◽  
pp. 41-86 ◽  
Author(s):  
Mark S. Joshi ◽  
Antônio Sá Barreto
Keyword(s):  
Inverse Scattering ◽  
Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic
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