Duke Mathematical Journal

Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Plamen Stefanov and Gunther Uhlmann

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Abstract

We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

Article information

Source
Duke Math. J. Volume 123, Number 3 (2004), 445-467.

Dates
First available in Project Euclid: 11 June 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1086957713

Digital Object Identifier
doi:10.1215/S0012-7094-04-12332-2

Mathematical Reviews number (MathSciNet)
MR2068966

Zentralblatt MATH identifier
1058.44003

Subjects
Primary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 53C24: Rigidity results 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 35R30: Inverse problems

Citation

Stefanov, Plamen; Uhlmann, Gunther. Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123 (2004), no. 3, 445--467. doi:10.1215/S0012-7094-04-12332-2. http://projecteuclid.org/euclid.dmj/1086957713.


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References

  • Yu. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl. 5 (1997), 487–480.
  • I. N. Bernstein and M. L. Gerver, “Conditions on distinguishability of metrics by hodographs (in Russian)” in Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 1980, 50–73.
  • E. Chappa, On the characterization of the kernel of the geodesic X-ray transform for tensor field, to appear in Trans. Amer. Math. Soc.
  • K. C. Creager, Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature 356 (1992), 309–314.
  • C. B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), 150–169.
  • –. –. –. –., Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), 445–464.
  • C. B. Croke, N. S. Dairbekov, and V. A. Sharafutdinov, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352 (2000), 3937–3956.
  • G. Eskin, Inverse scattering problem in anisotropic media, Comm. Math. Phys. 199 (1998), 471–491.
  • M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1–147.
  • V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977.
  • G. Herglotz, Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys. 52 (1905), 275–299.
  • M. Lassas, V. Sharafutdinov, and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann. 325 (2003), 767–793.
  • R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981), 71–83.
  • R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (in Russian), Dokl. Akad. Nauk SSSR 232, no. 1 (1977), 32–35.; English translation in Soviet Math. Dokl. 18, no. 1 (1977), 27–31.
  • –. –. –. –., On a problem of reconstructing Riemannian metrics, Siberian Math. J. 22, no. 3 (1981), 420–433.
  • R. G. Muhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an $n$-dimensional space (in Russian), Dokl. Akad. Nauk SSSR 243, no. 1 (1978), 41–44.
  • J.-P. Otal, Sur les longueur des géodésiques d'une métrique à coubure négative dans le disque, Comment. Math. Helv. 65 (1990), 334–347.
  • L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature (in Russian), Sibirsk. Mat. Zh. 29, no. 3 (1988), 114–130.; English translation in Siberian Math. J. 29, no. 3 (1988), 427–441.
  • V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser., VSP, Utrecht, Netherlands, 1994.
  • –. –. –. –., Finiteness theorem for the ray transform on a Riemannian manifold, Inverse Problems 11 (1995), 1039–1050.
  • V. Sharafutdinov and G. Uhlmann, On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, J. Differential Geom. 56 (2000), 93–110.
  • P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett. 5 (1998), 83–96.
  • M. E. Taylor, Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, 1981.
  • E. Wiechert and K. Zoeppritz, Über Erdbebenwellen, Koenigl. Nachr. Gesellsch. Wiss. Goettingen 4 (1907), 415–549.