Nowhere conformally homogeneous manifolds and limiting Carleman weights

Tony Liimatainen; Mikko Salo

doi:10.3934/ipi.2012.6.523

Article Contents

2012, Volume 6, Issue 3: 523-530. Doi: 10.3934/ipi.2012.6.523

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Nowhere conformally homogeneous manifolds and limiting Carleman weights

Tony Liimatainen^1, and
Mikko Salo^2,

1.
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100 FI-00076 Aalto
2.
Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, P.O. Box 35 FI-40014 Jyväskylä

Received: September 2010

Revised: July 2012

Published: August 2012

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Abstract

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analogue of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.

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Mathematics Subject Classification: Primary 35R30; Secondary 53A30.

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