American Institute of Mathematical Sciences

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

On the set of metrics without local limiting Carleman weights

Pablo Angulo-Ardoy

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.  Google Scholar

[2]

P. Angulo-Ardoy, D. Faraco and L. Guijarro, Sufficient Conditions for the Existence of Limiting Carleman Weights arXiv: 1603.04201 Google Scholar

[3]

J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry Springer, 1998.  Google Scholar

[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.  Google Scholar

[5]

M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[2]

P. Angulo-Ardoy, D. Faraco and L. Guijarro, Sufficient Conditions for the Existence of Limiting Carleman Weights arXiv: 1603.04201 Google Scholar

[3]

J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry Springer, 1998.  Google Scholar

[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.  Google Scholar

[5]

M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[1]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[2]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[3]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[4]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[5]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[6]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[7]

Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225
[8]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155
[9]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072
[10]

Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015
[11]

Annalisa Iuorio, Christian Kuehn, Peter Szmolyan. Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1269-1290. doi: 10.3934/dcdss.2020073
[12]

Tony Liimatainen, Mikko Salo. Nowhere conformally homogeneous manifolds and limiting Carleman weights. Inverse Problems & Imaging, 2012, 6 (3) : 523-530. doi: 10.3934/ipi.2012.6.523
[13]

Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
[14]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[15]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[16]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[17]

Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
[18]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[19]

Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
[20]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (59)
  • HTML views (132)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences