Finiteness of π1π1 and geometric inequalities in almost positive Ricci curvature

First eigenvalue of thep-Laplacian under integral curvature condition

“…Under these assumption, Aubry's diameter estimate implies that M is closed [3]. That paper also has the proof for p = 2.…”

“…We also need the following Sobolev inequality, which follows from Gallot's isoperimetric constant estimate for integral curvature [8] and Aubry's diameter estimate [3]. and K > 0, there exists ε = ε(n, q, K) such that if M n is a complete manifold with Ric…”

Diameter pinching in almost positive Ricci curvature

“…ball) of radius r in (S n , 1 k g). Besides the theorem 1.6, we will need the following comparison results for manifolds of almost positive Ricci curvature (see [4] for a proof).…”

“…In [4], we prove the following generalization of the Myers and Lichnerowicz theorems: Theorem 1.6. For any p> n/2, there exists C(p, n) such that if (M n , g) is a complete manifold with M Ric− (n−1)…”

“…is finite (for p > n/2) then Vol M is finite, and that we can not bound the diameter or the first non zero eigenvalue under the assumption ρ p ≤ 1 C(p,n) or ρ n 2 small (see [4]). …”