Inverse problems for Einstein manifolds

Colin Guillarmou; Antônio Sá Barreto

doi:10.3934/ipi.2009.3.1

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2009, Volume 3, Issue 1: 1-15. Doi: 10.3934/ipi.2009.3.1

This issue Previous Article Next Article A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination

Inverse problems for Einstein manifolds

Colin Guillarmou^1, and
Antônio Sá Barreto^2,

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

Received: December 31, 2007

Revised: October 31, 2008

Published: January 2009

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Abstract

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.

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Mathematics Subject Classification: Primary 58J50, Secondary 35P25.

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