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Rigidity for non-compact surfaces of finite area and certain Kähler manifolds

Published online by Cambridge University Press:  19 September 2008

Jianguo Cao
Jianguo Cao
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA (cao@math.Cornell.edu)
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Abstract

We first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 475 - 516
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[Ah]Ahlfors, L.. An extension of Schwarz's lemma. Trans. Amer. Math. Soc. 43 (1938), 359–364.Google Scholar
[Ba]Ballmann, W.. Axial isometries of manifolds of nonpositive curvature. Math. Ann. 259 (1982), 133–144.CrossRefGoogle Scholar
[BGS]Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Non-positive Curvature. Birkhaüser: Boston—Basel—Stüttgart, 1985.CrossRefGoogle Scholar
[Be1]Besse, A.. Manifolds All of Whose Geodesies are Closed. Springer: Berlin—Heidelberg—New York, 1978.CrossRefGoogle Scholar
[Be2]Besse, A.. Einstein Manifolds. Springer: Verlag, Berlin—Heidelberg—New York, 1987.CrossRefGoogle Scholar
[Br]Brooks, R.. The fundamental group and the spectrum of the Laplacian. Comm. Math. Helv. 56 (1981), 581–598.CrossRefGoogle Scholar
[Bo]Bonahon, F.. The geometry of Teichmuller space via geodesic currents Invent. Math. 92 (1988), 139–162.CrossRefGoogle Scholar
[BK]Burns, K. and Katok, A.. Manifolds with non-positive curvature. Ergod. Th. & Dynam. Sys. 5 (1985), 307–317.CrossRefGoogle Scholar
[Bu]Buser, P.. A note on the isoperimetric constant. Ann. Scient. Éc. Norm. Sup. 4⊃e série 16 (1985), 213–230.Google Scholar
[CaT]Carlson, J. A. and Toledo, D.. Harmonic mappings of Kähler manifolds to locally symmetric spaces. Publ. IHES 69 (1989), 173–201.CrossRefGoogle Scholar
[Ch]Cheeger, J.. A Lower Bound for the Smallest Eigenvalue of the Laplacian. In Gunning, R., ed. Problems in Analysis. (A Symposium in honor ofS. Böchner). Princeton University Press: Princeton, 1970. pp. 195–199.Google Scholar
[CE]Cheeger, J. and Ebin, D.. Comparison Theorems in Riemannian Geometry. North Holland: Amsterdam, 1975.Google Scholar
[Cn]Cheng, S. Y.. Eigenvalue comparison theorems and its applications. Math. Z. 143 (1975), 289–297.CrossRefGoogle Scholar
[CDP]Coornaet, M., Delzant, T. and Papadopoulos, A.. Géométrie et théorie des groupes. Les Groupes Hyperboliques de Gromov. Springer Lecture Notes in Mathematics 1441. Springer: Berlin, 1990.Google Scholar
[Cr]Croke, C.. Rigidity for surfaces of non-positive curvature. Comm. Math. Helv. 65 (1990), 150–169.CrossRefGoogle Scholar
[CFF]Croke, C., Fathi, A. and Feldman, J.. The marked length-spectrum of a surface of non-positive curvature. Topology 31 (1992), 847–855.CrossRefGoogle Scholar
[DR]Damek, E. and Ricci, F.. A class of non-symmetric harmonic Riemannian spaces. Bull. Amer. Math. Soc. 27 (1992), 139–142.CrossRefGoogle Scholar
[DoC]DoCarmo, M. P.. Differential Geometry of Curves and Surfaces. Prentice-Hall: (1976).Google Scholar
[E1]Eberlein, P.. Lattice in spaces of non-positive curvature. Ann. Math. 111 (1980), 435–476.CrossRefGoogle Scholar
[E2]Eberlein, P.. Geodesic flow in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
[E3]Eberlein, P.. Surfaces of non-positive curvature. Mem. Amer. Math. Soc. 218 (1979).Google Scholar
[E4]Eberlein, P.. Geodesic rigidity in compact non-positively curved manifolds. Trans. Amer. Math. Soc. 268 (1981), 411–443.CrossRefGoogle Scholar
[E5]Eberlein, P.. Geodesic flows on negatively curved manifolds 1. Ann. Math. 95 (1972), 492–510.CrossRefGoogle Scholar
[EO]Eberlein, P. and O'Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45–109.CrossRefGoogle Scholar
[ES]Eells, J. and Sampson, J. H.. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160.CrossRefGoogle Scholar
[Fa]Fathi, A.. Le spectre marqué des longueurs des surfaces sans points conjugués. C. R. Acad. Sci. Paris, Sér 1. Math. 309 (1989), 621–624.Google Scholar
[FJ]Farrell, T. and Jones, L.. Compact negatively curved manifolds (of dim ≠ 3, 4) are topologically rigid. Proc. Nat. Acad. Sci.. 86 (1989), 3461–3463.CrossRefGoogle ScholarPubMed
[FO]Feldman, J. and Ornstein, D.. Semi—rigidity of horocycle flows over surfaces of variable negative curvature. Ergod. Th. & Dynam. Sys. 7 (1987), 49–72.CrossRefGoogle Scholar
[FL]Foulon, P. and Labourie, F.. Sur les variétes compactes asymptotiquement harmoniques. Invent. Math. 109 (1991), 97–112.CrossRefGoogle Scholar
[Gr1]Gromov, M.. Three remarks on geodesic dynamic and fundamental groups info SUNY at Stony Brook. Preprint. (1976).Google Scholar
[Gr2]Gromov, M.. Manifolds of negative curvature. J. Diff. Geom. 13 (1978), 223–230.Google Scholar
[Gr3]Gromov, M.. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307–347.CrossRefGoogle Scholar
[Gr4]Gromov, M.. Hyperbolic Groups. In Gersten, M., eds. Essays in Group Theory. MSRI Series. Vol. 8. Springer: Berlin, 1987. pp. 75–265.CrossRefGoogle Scholar
[Gr5]Gromov, M.. Hyperbolic manifolds, groups and actions. In Kra, I. and Maskit, B., eds. Riemannian Surfaces and Related Topics. Annals of Mathematics Studies 97. Princeton University Press: Princeton, 1981. pp. 183–213.CrossRefGoogle Scholar
[Gr6]Gromov, M.. Hyperbolic Manifolds According to Thurston and Jorgensen. In Seminarie Bourbaki, 32 e annee. No. 546. (1979/1980), pp. 40–53.Google Scholar
[GK]Guillemin, V. and Kazhdan, D.. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301–312.CrossRefGoogle Scholar
[Ha]Hamenstädt, U.. Time preserving conjugacies of geodesic flows. Ergod. Th. & Dynam. Sys. 12 (1992), 67–74.CrossRefGoogle Scholar
[Ka]Katok, A.. Four applications of conformal equivalence to geometry and dynamics. Ergod. Th. & Dynam. Sys. 8* (1988), 139–152.Google Scholar
[Min]Min-Oo, M.. Spectral rigidity for manifolds with negative curvature operator. Contemp. Math., Nonlinear Problem in Geometry. 51 (1986), 99–103.Google Scholar
[Mo]Morse, M.. A fundamental class of geodesies on any closed surface of genus greater than one. Trans. Amer. Math. Soc. 26 (1924), 25–60.CrossRefGoogle Scholar
[Mu]Müller, W.. Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109 (1992), 265–305.CrossRefGoogle Scholar
[N]Nevanlinna, R.. Analytic Functions. Springer: Berlin, Heidelberg, 1970.CrossRefGoogle Scholar
[Ot1]Otal, J. P.. Le spectre marqué des longueurs des surfaces á courbure negative. Ann. Math. 131 (1990), 151–162.CrossRefGoogle Scholar
[Ot2]Otal, J. P.. Sur les longueurs des geodésiques dune metrique á courbure negative dans le disque. Comm. Math. Helv. 65 (1990), 334–347.CrossRefGoogle Scholar
[Pa]Pansu, P.. Quasi—conformal mappings and manifolds of negative curvature. Springer Lecture Notes in Mathematics 1201. 212–229.CrossRefGoogle Scholar
[Sa]Santaló, L. A.. Integral Geometry and Geometric Probability. Addison—Wesley: 1976.Google Scholar
[Sh]Short, H.. Notes on word hyperbolic groups. In Ghys, E., Haefliger, A. and Verjovsky, A., eds. Group Theory from a Geometrical Viewpoint. World Scientific: Singapore, Teaneck, New Jersey, 1991.Google Scholar
[Si]Siu, Y. T.. The complex-analyticity of harmonic maps and strong rigidity of compact Kahler manifolds. Ann. Math. 112 (1980), 73–111.CrossRefGoogle Scholar
[Sz]Szabo, Z. I.. Harmonic manifolds, The proof of the Lichnerowicz conjecture in compact case. J. Diff. Geom. 31 (1990), 1–28.Google Scholar
[Ya]Yang, C. T.. Any Blaschke manifold of homotopy type of ℂPn has the right volume. Pacific Math. J. 151 (1991), 379–394.CrossRefGoogle Scholar
[Y1]Yau, S. T.. A general Schwarz Lemma for Kähler manifolds. Amer. J. Math. 100 (1978), 197–203.CrossRefGoogle Scholar
[Y2]Yau, S. T.. Isoperimetric constants and the first eigenvalues of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Sup. 13 (1975), 487–507.CrossRefGoogle Scholar
[Y3]Yau, S. T.. Calabi's conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. 74 (1977), 1798–1799.CrossRefGoogle ScholarPubMed
[We]Weil, A.. Sur les surfaces a curbure negative. C. R. Acad. Sci. 182 (1926), 1069–1072.Google Scholar
[Wo]Wolpert, S.. The length spectra as moduli for compact Riemann surfaces. Ann. Math. 109 (1979), 323–351.CrossRefGoogle Scholar