DIAMETER PINCHING IN ALMOST POSITIVE RICCI CURVATURE
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DIAMETER PINCHING IN ALMOST POSITIVE RICCI CURVATURE ERWANN AUBRY Abstract. In this paper we prove a diameter sphere theorem and its corresponding ?1 sphere theorem under Lp control of the curvature. They are generalizations of some results due to S. Ilias [8]. 1. Introduction Let (Mn, g) be a complete manifold with Ricci curvature Ric ≥ n?1. Then (Mn, g) satisfies the following classical results (the proofs can be found in [13] for instance): • Diam(Mn, g) ≤ pi (S. Myers) with equality iff (Mn, g) = (Sn, can) (S. Cheng), • ?1(Mn, g) ≥ n (A. Lichnerowicz) with equality iff (Mn, g) = (Sn, can) (M. Obata), where Diam is the diameter and ?1 is the first positive eigenvalue. Studying the properties of the sphere kept by manifold with Ric ≥ n?1 and almost extremal diameter or ?1, S. Ilias proved in [8] the following results: Theorem 1.1 (S. Ilias). For any A> 0, there exists (A,n) > 0 such that any n-manifolds with Ric ≥ n?1, sectional curvature ? ≤ A and ?1 ≤ n+ is homeomorphic to Sn. Theorem 1.2 (S.
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DIAMETER PINCHING IN ALMOST POSITIVE RICCI CURVATURE
ERWANN AUBRY
Abstract.In this paper we prove a diameter sphere theorem and its corresponding p λ1sphere theorem underLcontrol of the curvature. They are generalizations of some results due to S. Ilias [8].
1.Introduction n n Let (M , g) be a complete manifold with Ricci curvature Ric≥n−(1. Then M , g) satisfies the following classical results (the proofs can be found in [13] for instance): n n n •Diam(M , g)≤π(S. Myers) with equality iff (M , g) = (S, can) (S. Cheng), n n n •λ1(gM , )≥n(A. Lichnerowicz) with equality iff (gM , ) = (S, can) (M. Obata), where Diam is the diameter andλ1is the first positive eigenvalue. Studying the properties of the sphere kept by manifold with Ric≥n−1 and almost extremal diameter orλ1, S. Ilias proved in [8] the following results: Theorem 1.1(S. Ilias).For anyA>0, there exists(A, n)>0such that anyn-manifolds n withRic≥n−1, sectional curvatureσ≤Aandλ1≤n+is homeomorphic toS. Theorem 1.2(S. Ilias).For anyA>0, there exists(A, n)>0such that anyn-manifolds n withRic≥n−1,σ≤AandDiam(M)≥π−is homeomorphic toS. Remark 1.3.C. Croke proves in[7]that forn-manifolds withRic≥n−1,λ1(M)close to nimpliesDiam(M)close toπ. The converse is proved in[8](using a spectral inequality due to S. Cheng[6]). Remark 1.4.Forn≥4, M. Anderson[1]and Y. Otsu[10]construct sequences of complete metricsgiwithRic(gi)≥n−1,λ1(gi)→nandDiam(gi)→πon manifolds that are not n homotope toS(more precisely, Otsu shows that ifn≥5, these manifolds can have infinitely many different fundamental groups). Remark 1.5.The two results of S. Ilias have been improved by G. Perelman in[11], where the assumptionσ≤Ais replaced byσ≥ −A(note that under the Ilias’s assumptions σ≤AandRic≥n−1we have|σ|≤(n−2)A). Subsequently, wedenoteRic(x)the lowest eigenvalue of the Ricci tensor andσ(x)the maximal sectional curvature atx.In [4], we prove the following generalization of the Myers and Lichnerowicz theorems: n Theorem 1.6.For anyp> n/2, there existsC(p, n)such that if(M , g)is a complete R p VolM manifold withRic−(n−1)<, thenMis compact, has finite fundamental M−C(p,n) group and satisfies h1i ρp 10 Diam(M)≤π1 +C(p, n) VolM h i 1 ρp p λ1(M)≥n1−C(p, n), VolM R p whereρp= Ric−(n−1)andx−= max(0,−x). M−
Key words and phrases.Ricci curvature, comparison theorems, integral bounds on the curvature, sphere theorems. 1
