Abstract
This paper is an exposition of some material from [P]. We explain how torsion in L p-cohomology can be used to prove a sharp pinching theorem for simply connected Riemannian manifolds with negative curvature. Namely, it is shown that a certain Riemannian homogeneous space whose curvature is negative and ¼-pinched cannot be quasi-isometric to any Riemannian manifold whose curvature is less than ¼-pinched.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Berger, De Sur certaines variétés riemanniennes à courbure positive. C. R. Acad. Sci., Paris 247, 1165–1168 (1958).
A. Borel, The L 2 -cohomology of negatively curved Riemannian symmetric spaces. Ann. Acad. Sci. Fennicae 10, 95–105 (1985).
J. Cheeger, M. Gromov, L 2 cohomology and group cohomology Topology 25, 189–215 (1986).
M. Chayet, N. Lohoue, Sur la cohomologie L p des variétés. C. R. Acad. Sci., Paris, Ser. I324, 211–213 (1997)
J. Dodziuk, L 2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Am. Math. Soc.77, 395–400 (197
H. Donnelly, F. Xavier, On the differential form spectrum for negatively curved manifolds. Amer. J. Math. 108, 169–185 (1984).
P. Eberlein, J. Heber, Quarter pinched homogeneous spaces of negative curvature. Internat. J. Math. 7, 441–500 (1996).
M. Gromov, Kähler hyperbolicity and L 2 -Hodge theory. J. Differential. Geom. 33, 253–320 (1991).
M. Gromov, Asymptotic invariants of infinite groups. In “Geometric Group Theory”, ed. G. Niblo and M. Roller, Cambridge University Press, Cambridge (1993).
V. Gol’dstein, M. Troyanov, The L p, q cohomology of Sol. Ann. Fac. Sci. Toulouse 7, 687–698 (1998).
U. Hamenstädt, Zur Theorie von Carnot-Caratheodory metriken und ihren Anwendungen. Bonner Math. Schriften 180, (1987).
E. Heintze, On homogeneous manifolds of negative curvature. Math. Annalen 211, 23–34 (1974).
J. Jost, Y. L. Xin, Vanishing theorems for L 2 -cohomology groups. J. Reine Angew. Math. 525, 95–112 2000.
L. Hernández Lamoneda, Kähler manifolds and 1/4 pinching. Duke Math. J. 62, 601–611 (1991).
A.N. Livsic, Cohomology of dynamical systems. Isv. Akad. Nauk SSSR, Ser. mat. 36, 1296–1320 (1972); translation from Math. USSR Izvestia 6, 1278-1301 (1972).
P. Pansu, Cohomologie L p, espaces homogènes et pincement. Prépublication d’Orsay (1999). http://www.math.u-psud.fr/∼pansu/
H. Triebel, Theory of function spaces IL Birkhaüser, Basel (1992).
M. Ville, On ¼-pinched 4-dimensional Riemannian manifolds of negative curvature. Ann. Global Anal. Geom. 3, 329–336 (1985).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pansu, P. (2002). L p-Cohomology and Pinching. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-662-04743-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07751-7
Online ISBN: 978-3-662-04743-9
eBook Packages: Springer Book Archive