On the set of metrics without local limiting Carleman weights

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January 2017, 11(1): 47-64. doi: 10.3934/ipi.2017003

On the set of metrics without local limiting Carleman weights

E.T.S de Ingenieros Navales, Universidad Politécnica de Madrid, Avd. Arco de la Victoria, No4, Ciudad Universitaria Madrid -28040, Spain

Received July 2015 Revised November 2016 Published January 2017

Fund Project: The author was supported by research grant ERC 301179

In the paper [1] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper [7] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of [1], in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is open and dense.

Keywords: Limiting Carleman weight, Calderón problem, transversality, Riemannian geometry, conformal geometry.

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

Citation: Pablo Angulo-Ardoy. On the set of metrics without local limiting Carleman weights. Inverse Problems & Imaging, 2017, 11 (1) : 47-64. doi: 10.3934/ipi.2017003

References:

[1]	P. Angulo-Ardoy, D. Faraco, L. Guijarro and A. Ruiz, Obstructions to the existence of limiting Carleman weights, Analysis & PDE, 9 (2016), 575-595. doi: 10.2140/apde.2016.9.575.
[2]	P. Angulo-Ardoy, D. Faraco and L. Guijarro, Sufficient Conditions for the Existence of Limiting Carleman Weights arXiv: 1603.04201
[3]	J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry Springer, 1998.
[4]	D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.
[5]	M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976.
[6]	S. Koike and M. Shiota, Non-smooth points set of fibres of a semialgebraic mapping, J. Math. Soc. Japan, 59 (2007), 953-969. doi: 10.2969/jmsj/05940953.
[7]	T. Liimatainen and M. Salo, Nowhere conformally homogeneous manifolds and limiting Carleman weights, Inverse Probl. Imaging, 6 (2012), 523-530. doi: 10.3934/ipi.2012.6.523.
[8]	C. T. C. Wall, Regular stratifications, Proceedings of "Applications of Topology and Dynamical Systems" at Warwick, Lecture Notes in Mathematics, 468 (1975), 332-345.

show all references

References:

[1]	P. Angulo-Ardoy, D. Faraco, L. Guijarro and A. Ruiz, Obstructions to the existence of limiting Carleman weights, Analysis & PDE, 9 (2016), 575-595. doi: 10.2140/apde.2016.9.575.
[2]	P. Angulo-Ardoy, D. Faraco and L. Guijarro, Sufficient Conditions for the Existence of Limiting Carleman Weights arXiv: 1603.04201
[3]	J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry Springer, 1998.
[4]	D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.
[5]	M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976.
[6]	S. Koike and M. Shiota, Non-smooth points set of fibres of a semialgebraic mapping, J. Math. Soc. Japan, 59 (2007), 953-969. doi: 10.2969/jmsj/05940953.
[7]	T. Liimatainen and M. Salo, Nowhere conformally homogeneous manifolds and limiting Carleman weights, Inverse Probl. Imaging, 6 (2012), 523-530. doi: 10.3934/ipi.2012.6.523.
[8]	C. T. C. Wall, Regular stratifications, Proceedings of "Applications of Topology and Dynamical Systems" at Warwick, Lecture Notes in Mathematics, 468 (1975), 332-345.

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