ON THE INTERPLAY BETWEEN INVERSE SCATTERING FOR ASYMPTOTICALLY HYPERBOLIC MANIFOLDS AND THE CALDERÓN PROBLEM FOR THE CONFORMAL LAPLACIAN

Abstract We use the relation obtained by Guillarmou and Guillopé [‘The determinant of the Dirichlet-to-Neumann map for surfaces with boundary’, Int. Math. Res. Not. IMRN 2007 (22) (2007), Article no. rnm099] and Chang and González [‘Fractional Laplacian in conformal geometry’, Adv. Math. 226 (2) (2011), 1410–1432], between the generalised eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension n + 1 $n+1$ n plus 1 with constant scalar curvature − n ( n + 1 ) $-n(n+1)$ minus n left parenthesis n plus 1 right parenthesis , the scattering matrix at energy ( n + 1 ) / 2 $(n+1)/2$ left parenthesis n plus 1 right parenthesis divided by 2 determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.