# American Institute of Mathematical Sciences

February  2009, 3(1): 1-15. doi: 10.3934/ipi.2009.3.1

## Inverse problems for Einstein manifolds

 1 Laboratoire J.-A. Dieudonné U.M.R. 6621 du C.N.R.S., Université de Nice Parc Valrose, 06108 Nice Cedex 02, France 2 Department of Mathematics Purdue University, 150 N. University Street, West-Lafayette IN 47907, United States

Received  January 2008 Revised  November 2008 Published  February 2009

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.
Citation: Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems and Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
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