Geometriae Dedicata

, Volume 164, Issue 1, pp 83–96 | Cite as

Local monotonicity of Riemannian and Finsler volume with respect to boundary distances

Original Paper


We show that the volume of a simple Riemannian metric on D n is locally monotone with respect to its boundary distance function. Namely if g is a simple metric on D n and g′ is sufficiently close to g and induces boundary distances greater or equal to those of g, then vol(D n , g′) ≥ vol(D n , g). Furthermore, the same holds for Finsler metrics and the Holmes–Thompson definition of volume. As an application, we give a new proof of injectivity of the geodesic ray transform for a simple Finsler metric.


Boundary distance function Minimal filling Finsler metric Holmes–Thompson volume Geodesic ray transform 

Mathematics Subject Classification (1991)

53C60 53C20 44A12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bao, D., Chern, S.-S., Shen, Z.: An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)Google Scholar
  2. 2.
    Bernstein I.N., Gerver M.L.: A problem of integral geometry for a family of geodesics and an inverse kinematic seismics problem (Russian). Dokl. Akad. Nauk SSSR 243(2), 302–305 (1978)MathSciNetGoogle Scholar
  3. 3.
    Bernstein I.N., Gerver M.L.: Conditions of Distinguishability of Metrics by Godographs (Russian), Methods and Algoritms of Interpretation of Seismological Information, Computerized Seismology, vol. 13, pp. 50–73. Nauka, Moscow (1980)Google Scholar
  4. 4.
    Besson G., Courtois G., Gallot S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Burago D., Ivanov S.: Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann. Math. (2) 171(2), 1183–1211 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burago, D., Ivanov S.: Area minimizers and boundary rigidity of almost hyperbolic metrics (2010) (preprint). arXiv:1011.1570 Google Scholar
  7. 7.
    Croke C.: Rigidity and the distance between boundary points. J. Differ. Geom. 33, 445–464 (1991)MathSciNetMATHGoogle Scholar
  8. 8.
    Croke C., Dairbekov N., Sharafutdinov V.: Local boundary rigidity of a compact Riemannian manifold with curvature bounded above. Trans. Am. Math. Soc 352(9), 3937–3956 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Croke C., Kleiner B.: A rigidity theorem for simply connected manifolds without conjugate points. Ergodic Theory Dyn. Syst. 18(4), 807–812 (1998)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gromov M.: Filling Riemannian manifolds. J. Differ. Geom. 18, 1–147 (1983)MathSciNetMATHGoogle Scholar
  11. 11.
    Holmes R.D., Thompson A.C.: N-dimensional area and content in Minkowski spaces. Pacific J. Math. 85, 77–110 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ivanov, S.: On two-dimensional minimal fillings, Algebra i Analiz 13(1), 26–38 (2001) (Russian) [St. Petersburg Math. J. 13, 17–25 (English) (2002)]Google Scholar
  13. 13.
    Ivanov, S.: Filling minimality of Finslerian 2-discs. Proc. Steklov Inst. Math. 273, 176–190 (2011). arXiv:0910.2257 Google Scholar
  14. 14.
    Ivanov, S.: Volume comparison via boundary distances. Proc. ICM 2, 769–784 (2010). arXiv:1004.2505 Google Scholar
  15. 15.
    Koehler, H.: On filling minimality of simple Finsler manifolds (2011) (preprint). arXiv:1107.1650v2 Google Scholar
  16. 16.
    Michel R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65, 71–83 (1981)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Mukhometov, R.G.: On a problem of integral geometry over geodesics of a Riemannian metric (Russian), Conditionally correct mathematical problems and problems of geophysics, Collect. Sci. Works, Novosibirsk 1979, pp. 86–125 (1979)Google Scholar
  18. 18.
    Muhometov, R.G.: On a problem of reconstructing Riemannian metrics. Sibirsk. Mat. Zh. 22(3), 119–135 (1981) (Russian) [Siberian Math. J. 22(3), 420–433 (1981) (English)]Google Scholar
  19. 19.
    Pestov L., Uhlmann G.: Two-dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. (2) 161, 1093–1110 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Santaló, L.A.: Integral geometry and geometric probability. Encyclopedia Math. Appl. Addison-Wesley, London (1976)Google Scholar
  21. 21.
    Sharafutdinov V. A.: An inverse problem of determining the source in the stationary transport equation for a Hamiltonian system. Sibirsk. Mat. Zh. 37(1), 211–235 (1996) (Russian) [Siberian Math. J. 37(1), 184–206 (1996) (English)]Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.St. Petersburg Department, V.A. Steklov Institute of MathematicsRussian Academy of SciencesSaint PetersburgRussia

Personalised recommendations