L p -Cohomology and Pinching

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 sim-ply connected Riemannian manifolds with negative curvature. Namely, it is shown that a certain Riemannian homogeneous space whose curvature is negative and 1 4 -pinched cannot be quasiisometric to any Riemannian manifold whose curvature is less than 1 4 -pinched. 1 Negative pinching Let −1 ≤ δ < 0. Say a Riemannian manifold is δ-pinched if its sectional curvature lies between −a and δa for some a > 0. 1.1 Examples : Rank one symmetric spaces of non compact type • Lobatchevski space H n R is −1-pinched. • Complex hyperbolic space H n C , n ≥ 2, quaternionic hyperbolic space H n H , n ≥ 2, and Cayley hyperbolic plane H 2 O are − 1 4 -pinched. 1.2 The motivating problem On a given manifold, what is the best possible pinching ? For simply con-nected (and thus non compact) manifolds, one must be more specific, and require that the unknown metric g be equivalent in the following sense to some reference metric g 0 : there exists C > 0 such that 1 C ≤ g g 0 ≤ C. We are fascinated by the following problem. Problem : On H n K (K = R, n ≥ 2), does there exist a metric which is δ-pinched for δ < − 1 4 and equivalent to the symmetric metric ? If one assumes the unknown metric to be periodic (i.e. admit a cocompact isometry group), then the answer is no, due to M. Ville [V] (dimension 4) and L. Hernández [Hz] (other cases). The general problem is open. Note that the corresponding problem for symmetric spaces of compact type has been solved by M. Berger and W. Klingenberg in 1958, [Be].