Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature

Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature Erwann AUBRY ? Abstra t We show that omplete n-manifolds whose part of Ri i urva- ture less than a positive number is small in Lp norm (for p > n/2) have bounded diameter and nite fundamental group. On the on- trary, omplete metri s with small Ln/2-norm of the same part of the Ri i urvature are dense in the set of metri s of any ompa t dierentiable manifold. Keywords: Ri i urvature, omparison theorems, fundamental group 1 Introdu tion A lassi al problem in Riemannian geometry is to nd topolog- i al, geometri al or analyti al ne essary onditions for the exis- ten e on a manifold of a Riemannian metri satisfying a given set of urvature bounds. For instan e, S. Myers showed that a omplete n-manifold with Ric≥k(n?1) (where k>0) is ompa t (the diameter is bounded by π√ k ) and has nite π1, whereas, on the ontrary, J. Lohkamp showed in [11? that on every n-manifold with n≥3 there exists a metri with negative Ri i urvature. This paper is devoted to the study of the Riemannian manifolds satis- fying only an Lp-pin hing on the negative lower part of their Ri i urvature tensors.

• eitherm has small


• stronger ur


• π1


• nite


• volume


• urvature bounded


• r0 ? π


• riemannian metri