Local boundary rigidity of a compact Riemannian manifold with curvature bounded above
HTML articles powered by AMS MathViewer
- by Christopher B. Croke, Nurlan S. Dairbekov and Vladimir A. Sharafutdinov PDF
- Trans. Amer. Math. Soc. 352 (2000), 3937-3956 Request permission
Abstract:
This paper considers the boundary rigidity problem for a compact convex Riemannian manifold $(M,g)$ with boundary $\partial M$ whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics $g’$ on $M$ there is a $C^{3,\alpha }$-neighborhood of $g$ such that $g$ is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance — as measured in $M$). More precisely, given any metric $g’$ in this neighborhood with the same boundary distance function there is diffeomorphism $\varphi$ which is the identity on $\partial M$ such that $g’=\varphi ^{*}g$. There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
- Michael T. Anderson, Remarks on the compactness of isospectral sets in low dimensions, Duke Math. J. 63 (1991), no. 3, 699–711. MR 1121151, DOI 10.1215/S0012-7094-91-06329-5
- Robert Brooks, Peter Perry, and Peter Petersen V, Compactness and finiteness theorems for isospectral manifolds, J. Reine Angew. Math. 426 (1992), 67–89. MR 1155747
- 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 for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150–169. MR 1036134, DOI 10.1007/BF02566599
- Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
- Christopher B. Croke and Vladimir A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology 37 (1998), no. 6, 1265–1273. MR 1632920, DOI 10.1016/S0040-9383(97)00086-4
- Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
- Victor Guillemin and David Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 153–180. MR 573432
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- 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
- Jean-Pierre Otal, Sur les longueurs des géodésiques d’une métrique à courbure négative dans le disque, Comment. Math. Helv. 65 (1990), no. 2, 334–347 (French). MR 1057248, DOI 10.1007/BF02566611
- V. A. Sharafutdinov, Integral geometry of a tensor field on a manifold with upper-bounded curvature, Sibirsk. Mat. Zh. 33 (1992), no. 3, 192–204, 221 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 3, 524–533 (1993). MR 1178471, DOI 10.1007/BF00970902
- 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, Some questions of integral geometry on Anosov manifolds, to appear.
- Luis A. SantalĂł, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
- Yu. A. Aminov and M. L. Rabelo, On toroidal submanifolds of constant negative curvature, Mat. Fiz. Anal. Geom. 2 (1995), no. 3-4, 275–283 (English, with English, Russian and Ukrainian summaries). MR 1484324
Additional Information
- Christopher B. Croke
- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- MR Author ID: 204906
- Email: ccroke@math.upenn.edu
- Nurlan S. Dairbekov
- Affiliation: Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia
- Email: dair@math.nsc.ru
- Vladimir A. Sharafutdinov
- Affiliation: Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia
- Email: sharaf@math.nsc.ru
- Received by editor(s): August 10, 1998
- Published electronically: May 22, 2000
- Additional Notes: The first and third authors were supported by CRDF, Grant RM2–143.
The first author was also supported by NSF grants #MDS-9505175 and #MDS-9626232.
The third author was also supported by INTAS — RFBR, Grant 95–0763. - © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 3937-3956
- MSC (2000): Primary 53C65, 53C22, 53C20; Secondary 58C25, 58C40
- DOI: https://doi.org/10.1090/S0002-9947-00-02532-0
- MathSciNet review: 1694283