Inverse problems for Einstein manifolds

We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1,$ we prove that the scattering matrix at energy $n$ on an open subset of its boundary determines the manifold up to isometries.

[1]  R. Melrose,et al.  Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature , 1987 .

[2]  John M. Lee,et al.  Determining anisotropic real-analytic conductivities by boundary measurements , 1989 .

[3]  Gunther Uhlmann,et al.  Developments in inverse problems since Calderon’s foundational paper , 1999 .

[4]  A. Nachman,et al.  Reconstructions from boundary measurements , 1988 .

[5]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[6]  Dennis DeTurck,et al.  Some regularity theorems in riemannian geometry , 1981 .

[7]  John M. Lee,et al.  Einstein metrics with prescribed conformal infinity on the ball , 1991 .

[8]  Melrose,et al.  Geometric Scattering Theory , 1995 .

[9]  Volume and Area Renormalizations for Conformally Compact Einstein Metrics , 1999, math/9909042.

[10]  M. S. Baouendi,et al.  UNIQUENESS FOR THE CHARACTERISTIC CAUCHY PROBLEM AND STRONG UNIQUE CONTINUATION FOR HIGHER ORDER PARTIAL DIFFERENTIAL INEQUALITIES , 1980 .

[11]  R. Mazzeo.  The Hodge cohomology of a conformally compact metric , 1988 .

[12]  Matti Lassas,et al.  The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary , 2003 .

[13]  A. Calderón,et al.  Uniqueness in the Cauchy Problem for Partial Differential Equations , 1958 .

[14]  Olivier Biquard,et al.  Reconstructing Minkowski space-time , 2004 .

[15]  Scattering matrix in conformal geometry , 2001, math/0109089.

[16]  Geometric aspects of the AdS/CFT correspondence , 2004, hep-th/0403087.

[17]  J. M. Thomas,et al.  Conformal Invariants. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Matti Lassas,et al.  On determining a Riemannian manifold from the Dirichlet-to-Neumann map , 2001 .

[19]  R. Mazzeo.  Unique Continuation at Infinity and Embedded Eigenvalues for Asymptotically Hyperbolic Manifolds , 1991 .

[20]  Olivier Biquard.  Continuation unique à partir de l'infini conforme pour les métriques d'Einstein , 2007, 0708.4346.

[21]  Mazzeo Rafe.  Elliptic theory of differential edge operators I , 1991 .

[22]  Antonio S'a Barreto,et al.  Inverse scattering on asymptotically hyperbolic manifolds , 1998, math/9811118.

[23]  C. Fefferman,et al.  The Ambient Metric , 2007, 0710.0919.