Duke Mathematical Journal

Area minimizers and boundary rigidity of almost hyperbolic metrics

Dmitri Burago and Sergei Ivanov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above-mentioned paper.

Article information

Duke Math. J., Volume 162, Number 7 (2013), 1205-1248.

First available in Project Euclid: 10 May 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C24: Rigidity results
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]


Burago, Dmitri; Ivanov, Sergei. Area minimizers and boundary rigidity of almost hyperbolic metrics. Duke Math. J. 162 (2013), no. 7, 1205--1248. doi:10.1215/00127094-2142529. https://projecteuclid.org/euclid.dmj/1368193652

Export citation


  • [1] G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731–799.
  • [2] D. Burago and S. Ivanov, On asymptotic volume of tori, Geom. Funct. Anal. 5 (1995), 800–808.
  • [3] D. Burago and S. Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. (2) 156 (2002), 891–914.
  • [4] D. Burago and S. Ivanov, Gaussian images of surfaces and ellipticity of surface area functionals, Geom. Funct. Anal. 14 (2004), 469–490.
  • [5] D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Ann. of Math. (2) 171 (2010), 1183–1211.
  • [6] C. B. Croke, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), 445–464.
  • [7] C. B. Croke, “Rigidity theorems in Riemannian geometry” in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl. 137, Springer, New York, 2004, 47–72.
  • [8] C. B. Croke and B. Kleiner, A rigidity theorem for simply connected manifolds without conjugate points, Ergodic Theory Dynam. Systems 18 (1998), 807–812.
  • [9] M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1–147.
  • [10] R. D. Holmes and A. C. Thompson, $n$-dimensional area and content in Minkowski spaces, Pacific J. Math. 85 (1979), 77–110.
  • [11] S. V. Ivanov, On two-dimensional minimal fillings (in Russian), Algebra i Analiz 13 (2001), no. 1, 26–38; English translation in St. Petersburg Math. J. 13 (2002), 17–25.
  • [12] S. Ivanov, Volumes and areas of Lipschitz metrics (in Russian), Algebra i Analiz 20 (2008), no. 3, 74–111; English translation in St. Petersburg Math. J. 20 (2009), 381–405.
  • [13] S. Ivanov, “Volume comparison via boundary distances” in Proceedings of the International Congress of Mathematicians, II, Hindustan Book Agency, New Delhi, 2010, 769–784.
  • [14] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), 71–83.
  • [15] L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2) 161 (2005), 1093–1110.
  • [16] L. A. Santaló, Integral Geometry and Geometric Probability, Encyclopedia Math. Appl. 1, Addison-Wesley, Reading, Mass., 1976.