Skip to main content Accessibility help
×
Home

Marked boundary rigidity for surfaces

  • COLIN GUILLARMOU (a1) and MARCO MAZZUCCHELLI (a2)

Abstract

We show that, on an oriented compact surface, two sufficiently $C^{2}$ -close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

Copyright

References

Hide All
[CFF92] Croke, C. B., Fathi, A. and Feldman, J.. The marked length-spectrum of a surface of nonpositive curvature. Topology 31(4) (1992), 847855.
[CH11] Croke, C. B. and Herreros, P.. Lens rigidity with trapped geodesics in two dimensions. Asian J. Math. 20(1) (2016), 4757.
[Cro90] Croke, C. B.. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1) (1990), 150169.
[Cro91] Croke, C. B.. Rigidity and the distance between boundary points. J. Differential Geom. 33(2) (1991), 445464.
[DG14a] Dyatlov, S. and Guillarmou, C.. Microlocal limits of plane waves and Eisenstein functions. Ann. Sci. Éc. Norm. Supér. (4) 47(2) (2014), 371448.
[DG14b] Dyatlov, S. and Guillarmou, C.. Pollicott–Ruelle resonances for open systems. Ann. Henri Poincaré (2016) to appear, Preprint arXiv:1410.5516.
[ES70] Earle, C. J. and Schatz, A.. Teichmüller theory for surfaces with boundary. J. Differential Geom. 4 (1970), 169185.
[GKM75] Gromoll, D., Klingenberg, W. and Meyer, W.. Riemannsche Geometrie im Großen (Lecture Notes in Mathematics, 55) . Springer, Berlin–New York, 1975, Zweite Auflage.
[Gui14] Guillarmou, C.. Lens rigidity for manifolds with hyperbolic trapped set. J. Amer. Math. Soc. (2014), to appear, arXiv:1412.1760.
[Hof85] Hofer, H.. A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. Lond. Math. Soc. (2) 31(3) (1985), 566570.
[KH95] Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.
[Mic82] Michel, R.. Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65(1) (1981/82), 7183.
[Mil63] Milnor, J.. Morse theory. Based on Lecture Notes by M. Spivak and R. Wells (Annals of Mathematics Studies, 51) . Princeton University Press, Princeton, NJ, 1963.
[Muk81] Mukhometov, R. G.. On a problem of reconstructing Riemannian metrics. Sibirsk. Mat. Zh. 22(3) (1981), 119135, 237.
[Ota90a] Otal, J.-P.. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2) 131(1) (1990), 151162.
[Ota90b] Otal, J.-P.. Sur les longueurs des géodésiques d’une métrique à courbure négative dans le disque. Comment. Math. Helv. 65(2) (1990), 334347.
[Pat99] Paternain, G. P.. Geodesic Flows (Progress in Mathematics, 180) . Birkhäuser Boston, Boston, MA, 1999.
[PU05] Pestov, L. and Uhlmann, G.. Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. of Math. (2) 161(2) (2005), 10931110.

Related content

Powered by UNSILO

Marked boundary rigidity for surfaces

  • COLIN GUILLARMOU (a1) and MARCO MAZZUCCHELLI (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.