## Boundary rigidity with partial data

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- by Plamen Stefanov, Gunther Uhlmann and Andras Vasy
- J. Amer. Math. Soc.
**29**(2016), 299-332 - DOI: https://doi.org/10.1090/jams/846
- Published electronically: November 17, 2015
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## Abstract:

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary.

The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces.

## References

- Gang Bao and Hai Zhang,
*Sensitivity analysis of an inverse problem for the wave equation with caustics*, J. Amer. Math. Soc.**27**(2014), no. 4, 953–981. MR**3230816**, DOI 10.1090/S0894-0347-2014-00787-6 - G. Besson, G. Courtois, and S. Gallot,
*Entropies et rigidités des espaces localement symétriques de courbure strictement négative*, Geom. Funct. Anal.**5**(1995), no. 5, 731–799 (French). MR**1354289**, DOI 10.1007/BF01897050 - Dmitri Burago and Sergei Ivanov,
*Boundary rigidity and filling volume minimality of metrics close to a flat one*, Ann. of Math. (2)**171**(2010), no. 2, 1183–1211. MR**2630062**, DOI 10.4007/annals.2010.171.1183 - Christopher Croke,
*Scattering rigidity with trapped geodesics*, Ergodic Theory Dynam. Systems**34**(2014), no. 3, 826–836. MR**3199795**, DOI 10.1017/etds.2012.164 - K. C. Creager,
*Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK*, Nature**356**(1992), 309–314., DOI 10.1038/356309a0 - Christopher B. Croke,
*Rigidity for surfaces of nonpositive curvature*, Comment. Math. Helv.**65**(1990), no. 1, 150–169. MR**1036134**, DOI 10.1007/BF02566599 - Christopher B. Croke,
*Rigidity and the distance between boundary points*, J. Differential Geom.**33**(1991), no. 2, 445–464. MR**1094465** - Christopher B. Croke,
*Rigidity theorems in Riemannian geometry*, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 47–72. MR**2169902**, DOI 10.1007/978-1-4684-9375-7_{4} - Christopher B. Croke and Nurlan S. Dairbekov,
*Lengths and volumes in Riemannian manifolds*, Duke Math. J.**125**(2004), no. 1, 1–14. MR**2097355**, DOI 10.1215/S0012-7094-04-12511-4 - Christopher B. Croke and Bruce Kleiner,
*Conjugacy and rigidity for manifolds with a parallel vector field*, J. Differential Geom.**39**(1994), no. 3, 659–680. MR**1274134** - Nurlan S. Dairbekov,
*Integral geometry problem for nontrapping manifolds*, Inverse Problems**22**(2006), no. 2, 431–445. MR**2216407**, DOI 10.1088/0266-5611/22/2/003 - C. Guillarmou,
*Lens rigidity for manifolds with hyperbolic trapped set*, available at arXiv:1412.1760., DOI 10.1090/jams/865 - Mikhael Gromov,
*Filling Riemannian manifolds*, J. Differential Geom.**18**(1983), no. 1, 1–147. MR**697984** - G. Herglotz,
*Über die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte*, Z. Math. Phys.**52**(1905), 275–299. - L. Hörmander,
*The analysis of linear partial differential operators*, Vol. 1–4, Springer-Verlag, Berlin, 1983. - Sergei Ivanov,
*Volume comparison via boundary distances*, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 769–784. MR**2827818** - Venkateswaran P. Krishnan,
*A support theorem for the geodesic ray transform on functions*, J. Fourier Anal. Appl.**15**(2009), no. 4, 515–520. MR**2549942**, DOI 10.1007/s00041-009-9061-5 - Venkateswaran P. Krishnan and Plamen Stefanov,
*A support theorem for the geodesic ray transform of symmetric tensor fields*, Inverse Probl. Imaging**3**(2009), no. 3, 453–464. MR**2557914**, DOI 10.3934/ipi.2009.3.453 - Matti Lassas, Vladimir Sharafutdinov, and Gunther Uhlmann,
*Semiglobal boundary rigidity for Riemannian metrics*, Math. Ann.**325**(2003), no. 4, 767–793. MR**1974568**, DOI 10.1007/s00208-002-0407-4 - Richard B. Melrose,
*Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces*, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 85–130. MR**1291640** - René Michel,
*Sur la rigidité imposée par la longueur des géodésiques*, Invent. Math.**65**(1981/82), no. 1, 71–83 (French). MR**636880**, DOI 10.1007/BF01389295 - R. G. Muhometov,
*The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry*, Dokl. Akad. Nauk SSSR**232**(1977), no. 1, 32–35 (Russian). MR**0431074** - R. G. Muhometov,
*On a problem of reconstructing Riemannian metrics*, Sibirsk. Mat. Zh.**22**(1981), no. 3, 119–135, 237 (Russian). MR**621466** - R. G. Muhometov 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). MR**511273** - Cesare Parenti,
*Operatori pseudo-differenziali in $R^{n}$ e applicazioni*, Ann. Mat. Pura Appl. (4)**93**(1972), 359–389. MR**437917**, DOI 10.1007/BF02412028 - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*Tensor tomography on surfaces*, Invent. Math.**193**(2013), no. 1, 229–247. MR**3069117**, DOI 10.1007/s00222-012-0432-1 - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*The attenuated ray transform for connections and Higgs fields*, Geom. Funct. Anal.**22**(2012), no. 5, 1460–1489. MR**2989440**, DOI 10.1007/s00039-012-0183-6 - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*Tensor tomography: progress and challenges*, Chinese Ann. Math. Ser. B**35**(2014), no. 3, 399–428. MR**3200025**, DOI 10.1007/s11401-014-0834-z - Leonid Pestov and Gunther Uhlmann,
*Two dimensional compact simple Riemannian manifolds are boundary distance rigid*, Ann. of Math. (2)**161**(2005), no. 2, 1093–1110. MR**2153407**, DOI 10.4007/annals.2005.161.1093 - L. N. Pestov and V. A. Sharafutdinov,
*Integral geometry of tensor fields on a manifold of negative curvature*, Sibirsk. Mat. Zh.**29**(1988), no. 3, 114–130, 221 (Russian); English transl., Siberian Math. J.**29**(1988), no. 3, 427–441 (1989). MR**953028**, DOI 10.1007/BF00969652 - Akhil Ranjan and Hemangi Shah,
*Convexity of spheres in a manifold without conjugate points*, Proc. Indian Acad. Sci. Math. Sci.**112**(2002), no. 4, 595–599. MR**1941895**, DOI 10.1007/BF02829692 - Vladimir Sharafutdinov, Michal Skokan, and Gunther Uhlmann,
*Regularity of ghosts in tensor tomography*, J. Geom. Anal.**15**(2005), no. 3, 499–542. MR**2190243**, DOI 10.1007/BF02930983 - V. A. Sharafutdinov,
*Integral geometry of tensor fields*, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR**1374572**, DOI 10.1515/9783110900095 - V. A. Sharafutdinov,
*Integral geometry of a tensor field on a surface of revolution*, Sibirsk. Mat. Zh.**38**(1997), no. 3, 697–714, iv (Russian, with Russian summary); English transl., Siberian Math. J.**38**(1997), no. 3, 603–620. MR**1457488**, DOI 10.1007/BF02683847 - V. A. Sharafutdinov,
*A problem in integral geometry in a nonconvex domain*, Sibirsk. Mat. Zh.**43**(2002), no. 6, 1430–1442 (Russian, with Russian summary); English transl., Siberian Math. J.**43**(2002), no. 6, 1159–1168. MR**1946241**, DOI 10.1023/A:1021189922555 - Vladimir Sharafutdinov,
*Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds*, J. Geom. Anal.**17**(2007), no. 1, 147–187. MR**2302878**, DOI 10.1007/BF02922087 - M. A. Šubin,
*Pseudodifferential operators in $R^{n}$*, Dokl. Akad. Nauk SSSR**196**(1971), 316–319 (Russian). MR**0273463** - Plamen Stefanov,
*Microlocal approach to tensor tomography and boundary and lens rigidity*, Serdica Math. J.**34**(2008), no. 1, 67–112. MR**2414415** - Plamen Stefanov and Gunther Uhlmann,
*Rigidity for metrics with the same lengths of geodesics*, Math. Res. Lett.**5**(1998), no. 1-2, 83–96. MR**1618347**, DOI 10.4310/MRL.1998.v5.n1.a7 - 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 10.1215/S0012-7094-04-12332-2 - Plamen Stefanov and Gunther Uhlmann,
*Boundary rigidity and stability for generic simple metrics*, J. Amer. Math. Soc.**18**(2005), no. 4, 975–1003. MR**2163868**, DOI 10.1090/S0894-0347-05-00494-7 - Plamen Stefanov and Gunther Uhlmann,
*Boundary and lens rigidity, tensor tomography and analytic microlocal analysis*, Algebraic analysis of differential equations from microlocal analysis to exponential asymptotics, Springer, Tokyo, 2008, pp. 275–293. MR**2758914**, DOI 10.1007/978-4-431-73240-2_{2}3 - Plamen Stefanov and Gunther Uhlmann,
*Integral geometry on tensor fields on a class of non-simple Riemannian manifolds*, Amer. J. Math.**130**(2008), no. 1, 239–268. MR**2382148**, DOI 10.1353/ajm.2008.0003 - Plamen Stefanov and Gunther Uhlmann,
*Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds*, J. Differential Geom.**82**(2009), no. 2, 383–409. MR**2520797** - Plamen Stefanov and Gunther Uhlmann,
*Recovery of a source term or a speed with one measurement and applications*, Trans. Amer. Math. Soc.**365**(2013), no. 11, 5737–5758. MR**3091263**, DOI 10.1090/S0002-9947-2013-05703-0 - Plamen Stefanov and Gunther Uhlmann,
*The geodesic X-ray transform with fold caustics*, Anal. PDE**5**(2012), no. 2, 219–260. MR**2970707**, DOI 10.2140/apde.2012.5.219 - P. Stefanov and G. Uhlmann,
*Multi-wave methods via ultrasound*, in Inside Out, Vol. 60, MSRI Publications, Cambridge, UK, 2009. - Hans Triebel,
*Interpolation theory, function spaces, differential operators*, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR**503903** - G. Uhlmann and A. Vasy,
*The inverse problem for the local geodesic ray transform*, available at arXiv:1210.2084. To appear in Invent. Math., DOI 10.1007/s00222-015-0631-7 - James Vargo,
*A proof of lens rigidity in the category of analytic metrics*, Math. Res. Lett.**16**(2009), no. 6, 1057–1069. MR**2576693**, DOI 10.4310/MRL.2009.v16.n6.a13 - E. Wiechert and K. Zoeppritz,
*Über Erdbebenwellen*, Nachr. Koenigl. Geselschaft Wiss. Göttingen**4**(1907), 415–549. - H. Zhou,
*The inverse problem for the local ray transform*, available at arXiv:1304.7023.

## Bibliographic Information

**Plamen Stefanov**- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 166695
- Email: Plamen-Stefanov@purdue.edu
**Gunther Uhlmann**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195 and Department of Mathematics, University of Helsinki, Finland FI-00014
- MR Author ID: 175790
- Email: gunther@math.washington.edu
**Andras Vasy**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 616271
- Email: andras@stanford.edu
- Received by editor(s): November 13, 2013
- Published electronically: November 17, 2015
- Additional Notes: The first author was partly supported by NSF Grant DMS-1301646

The second author was partly supported by NSF Grants CMG-1025259 and DMS-1265958, The Fondation Mathématiques de Paris, and a Simons fellowship

The third author was partly supported by NSF Grants CMG-1025259 and DMS-1068742. - © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 299-332 - MSC (2010): Primary 53C24, 35R30
- DOI: https://doi.org/10.1090/jams/846
- MathSciNet review: 3454376