Metric tensor estimates, geometric convergence, and inverse boundary problems
Authors:
Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 69-79
MSC (2000):
Primary 35J25, 47A52, 53C21
DOI:
https://doi.org/10.1090/S1079-6762-03-00113-6
Published electronically:
September 2, 2003
MathSciNet review:
2029467
Full-text PDF Free Access
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Abstract: Three themes are treated in the results announced here. The first is the regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is the geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel’fand, making essential use of the results of the first two parts.
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[An2]An2 M. Anderson, Boundary regularity, uniqueness, and non-uniqueness for AH Einstein metrics on 4-manifolds, Advances in Math., to appear.
[AK2LT]AK2LT M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem, Preprint, 2002.
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[K2L]K2L A. Katsuda, Y. Kurylev, and M. Lassas, Stability in Gelfand inverse boundary spectral problem, Preprint, 2001.
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[An1]An1 M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445.
[An2]An2 M. Anderson, Boundary regularity, uniqueness, and non-uniqueness for AH Einstein metrics on 4-manifolds, Advances in Math., to appear.
[AK2LT]AK2LT M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem, Preprint, 2002.
[Be1]Be1 M. Belishev, An approach to multidimensional inverse problems, Dokl. Akad. Nauk. SSSR 297 (1987), 524–527; English transl., Soviet Math. Dokl. 36 (1988), no. 3, 481–484.
[BK1]BK1 M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. PDE 17 (1992), 767–804.
[Ch]Ch J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–75.
[CG]CG J. Cheeger and D. Gromoll, The splitting theorem for manifolds with non-negative Ricci curvature, J. Diff. Geom. 6 (1971), 119–128.
[DTK]DTK D. De Turck and J. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Scient. Ecole Norm. Sup. Paris 14 (1981), 249–260.
[Ge]Ge I. Gel’fand, Some aspects of functional analysis and algebra, Proc. ICM 1 (1954), 253–277.
[Gr]Gr M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces (with appendices by M. Katz, P. Pansu, and S. Semmes), Birkhäuser, Boston, 1999.
[HH]HH M. Hebey and M. Herzlich, Harmonic coordinates, harmonic radius, and convergence of Riemannian manifolds, Rend. di Mathem. Ser. VII, 17 (1997), 569-605.
[KKL]KKL A. Katchalov, Y. Kurylev, and M. Lassas, Inverse Boundary Spectral Problems, Chapman-Hall/CRC Press, Boca Raton, 2001.
[K]K T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
[Ka]Ka A. Katsuda, BC-method and stability of Gel’fand inverse spectral problem, Proc. Conf. “Spectral and Scattering Theory,” RIMS, Kyoto, 2001, 24–35.
[K2L]K2L A. Katsuda, Y. Kurylev, and M. Lassas, Stability in Gelfand inverse boundary spectral problem, Preprint, 2001.
[Ku]Ku Y. Kurylev, Multidimensional Gel’fand inverse boundary problem and boundary distance map. In: Inverse Problems Related to Geometry (H. Soga, ed.), 1–15, Ibaraki Univ. Press, Mito, 1997.
[KuL]KuL Y. Kurylev and M. Lassas, The multidimensional Gel’fand inverse problem for non-self-adjoint operators, Inverse Problems 13 (1997), 1495–1501.
[Pe]Pe P. Petersen, Riemannian Geometry, Springer-Verlag, New York, 1998.
[Ta]Ta D. Tataru, Unique continuation for solutions to PDE’s: between Hörmander’s theorem and Holmgren’s theorem, Comm. PDE 20 (1995), 855–884.
[T2]T2 M. Taylor, Tools for PDE, AMS, Providence, RI, 2000.
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Additional Information
Michael Anderson
Affiliation:
Mathematics Department, State University of New York, Stony Brook, NY 11794
Email:
anderson@math.sunysb.edu
Atsushi Katsuda
Affiliation:
Mathematics Department, Okayama University, Tsushima-naka, Okayama, 700-8530, Japan
MR Author ID:
227490
Email:
katsuda@math.okayama-u.ac.jp
Yaroslav Kurylev
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK
Email:
Y.V.Kurylev@lboro.ac.uk
Matti Lassas
Affiliation:
Rolf Nevanlinna Institute, University of Helsinki, FIN-00014, Finland
Email:
lassas@cc.helsinki.fi
Michael Taylor
Affiliation:
Mathematics Deptartment, University of North Carolina, Chapel Hill, NC 27599
MR Author ID:
210423
Email:
met@math.unc.edu
Keywords:
Ricci tensor,
harmonic coordinates,
geometric convergence,
inverse problems,
conditional stability
Received by editor(s):
December 17, 2002
Published electronically:
September 2, 2003
Communicated by:
Tobias Colding
Article copyright:
© Copyright 2003
American Mathematical Society
