Two conformal asymptotically hyperbolic metrics that are simple or negatively curved are identical if their marked renormalized boundary distances coincide for some choices of conformal representatives.
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Boundary Rigidity in a fixed conformal class for Asymptotically Hyperbolic Manifolds
Two conformal asymptotically hyperbolic metrics that are simple or negatively curved are identical if their marked renormalized boundary distances coincide for some choices of conformal representatives.