# Finiteness of $\pi_1$ and geometric inequalities in almost positive Ricci curvature.

Abstract : We show that complete $n$-manifolds whose part of Ricci curvature less than a positive number is small in $L^p$ norm (for $p>n/2$) have bounded diameter and finite fundamental group. On the contrary, complete metrics with small $L^{n/2}$-norm of the same part of the Ricci curvature are dense in the set of metrics of any compact differentiable manifold.
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Erwann Aubry. Finiteness of $\pi_1$ and geometric inequalities in almost positive Ricci curvature.. Annales Scientifiques de l'École Normale Supérieure, Société mathématique de France, 2007, 40 (4), pp.675-695. ⟨hal-00338175⟩

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