Recent progress on the boundary rigidity problem
Authors:
Plamen Stefanov and Gunther Uhlmann
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 64-70
MSC (2000):
Primary 53C20
DOI:
https://doi.org/10.1090/S1079-6762-05-00148-4
Published electronically:
June 23, 2005
MathSciNet review:
2150946
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Abstract: The boundary rigidity problem consists in determining a compact, Riemannian manifold with boundary, up to isometry, by knowing the boundary distance function between boundary points. In this paper we announce the result of our forthcoming article that one can solve this problem for generic simple metrics. Moreover we probe stability estimates for this problem.
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- Plamen Stefanov and Gunther Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (2004), no. 3, 445–467. MR 2068966, DOI https://doi.org/10.1215/S0012-7094-04-12332-2
SU3 P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, IMRN 17 (2005), 1047–1061.
SU4 P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, preprint, arXiv:math.DG/0408075.
- Daniel Tataru, Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations 20 (1995), no. 5-6, 855–884. MR 1326909, DOI https://doi.org/10.1080/03605309508821117
- François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York-London, 1980. Pseudodifferential operators; The University Series in Mathematics. MR 597144
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WZ E. Wiechert E and K. Zoeppritz, Über Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss. Göttingen 4 (1907), 415–549.
BK M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its boundary spectral data (BC-method), Comm. PDE 17 (1992), 767–804.
BG I. N. Bernstein and M. L. Gerver, Conditions on distinguishability of metrics by hodographs, Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, pp. 50–73. (Russian)
BCGG. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731–799.
B G. Beylkin, Stability and uniqueness of the solution of the inverse kinematic problem in the multidimensional case, J. Soviet Math. 21 (1983), 251–254.
BI D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, manuscript, 2005.
Cr K. C. Creager, Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature 356 (1992), 309–314.
C2 C. Croke, Rigidity for surfaces of non-positive curvature, Comment. Math. Helv. 65 (1990), 150–169.
C1 C. Croke, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), no. 2, 445–464.
CDS C. Croke, N. Dairbekov, and V. Sharafutdinov, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352 (2000), no. 9, 3937–3956.
Gr M. Gromov, Filling Riemannian manifolds, J. Diff. Geometry 18 (1983), no. 1, 1–148.
H G. Herglotz, Über die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Zeitschr. für Math. Phys. 52 (1905), 275–299.
LSU M. Lassas, V. Sharafutdinov, and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann. 325 (2003), 767–793.
M R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981), 71–83.
Mu R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR 232 (1977), no. 1, 32–35. (Russian)
Mu2 R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, Siberian Math. J. 22 (1982), no. 3, 420–433.
Mu-R R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an $n$-dimensional space, Dokl. Akad. Nauk SSSR 243 (1978), no. 1, 41–44. (Russian)
O J. P. Otal, Sur les longuers des géodésiques d’une métrique à courbure négative dans le disque, Comment. Math. Helv. 65 (1990), 334–347.
PS L. Pestov and V. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature Sibirsk. Mat. Zh. 29 (1988), no. 3, 114–130; English transl., Siberian Math. J. 29 (1988), no. 3, 427–441.
PU L. Pestov and G. Uhlmann, Two-dimensional simple compact manifolds with boundary are boundary rigid, to appear in Annals of Math.
Sh V. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht, 1994.
SSU V. Sharafutdinov, M. Skokan, and G. Uhlmann, Regularity of ghosts in tensor tomography, preprint.
SU V. Sharafutdinov and G. Uhlmann, On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, Journal of Differential Geometry 56 (2001), 93–110.
SU1 P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett. 8 (2001), no. 1-2, 105–124.
SU2 P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (2004), 445–467.
SU3 P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, IMRN 17 (2005), 1047–1061.
SU4 P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, preprint, arXiv:math.DG/0408075.
Ta D. Tataru, Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Comm. P.D.E. 20 (1995), 855–884.
T F. Trèves, Introduction to pseudodifferential and Fourier integral operators, Vol. 1. Pseudodifferential operators, The University Series in Mathematics, Plenum Press, New York–London, 1980.
W J. Wang, Stability for the reconstruction of a Riemannian metric by boundary measurements, Inverse Probl. 15 (1999), 1177–1192.
WZ E. Wiechert E and K. Zoeppritz, Über Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss. Göttingen 4 (1907), 415–549.
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Additional Information
Plamen Stefanov
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907
MR Author ID:
166695
Email:
stefanov@math.purdue.edu
Gunther Uhlmann
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195
MR Author ID:
175790
Email:
gunther@math.washington.edu
Keywords:
Boundary rigidity,
Riemannian manifold,
inverse problem
Received by editor(s):
March 8, 2005
Published electronically:
June 23, 2005
Additional Notes:
The first author was supported in part by NSF Grant DMS-0400869.
The second author was supported in part by NSF Grant DMS-0245414.
Communicated by:
Dmitri Burago
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
