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.

• exists x0


• hence ? ?


• ?f then


• since ∫


• pi ?


• ?p ≤


• linear connection


• there exists

Sujets

• Anderson
• Linear connection
• Existential quantification

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

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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 λ 1 sphere 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 •λ 1 (gM , )≥n(A. Lichnerowicz) with equality iff (gM , ) = (S, can) (M. Obata), where Diam is the diameter andλ 1 is the first positive eigenvalue. Studying the properties of the sphere kept by manifold with Ric≥n−1 and almost extremal diameter orλ 1 , 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≤Aandλ 1 ≤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,λ 1 (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 metricsg i withRic(g i )≥n−1,λ 1 (g i )→nandDiam(g i )→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 h 1 i   ρ p 10 Diam(M)≤π1 +C(p, n) VolM h i   1 ρ p p λ 1 (M)≥n1−C(p, n), VolM R   p whereρ p = Ric−(n−1)andx − = max(0,−x). M −

Key words and phrases.Ricci curvature, comparison theorems, integral bounds on the curvature, sphere theorems. 1