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Project Description Project Description Gromov-Hausdorff convergence has been employed with great success in the past decade to study a number of almost rigidity conjectures regarding manifolds with lower bounds on their Ricci curvature. Increasingly mathematicians have become interested in the geometric and topological properties of the limit spaces of these manifolds. Recall that such limit spaces were shown to be complete length spaces by Gromov [Gr]. That is, they are metric spaces such that every pair of points is joined by a length minimizing curve. One of the many equivalent definitions of Gromov-Hausdorff distance is given using the notion of a correspondance. Definition 1. Given two metric spaces, X and Y , a correspondance is a set R ∈ X × Y such that ∀x ∈ X, ∃y ∈ Y such that (x, y) ∈ R and visa versa. [BBI] Definition 2. The distortion of R is defined by 0 0 0 0 (1) dis(R) = sup{|dX (x, x ) − dY (y, y ):(x, y), (x , y ) ∈ R}. Then the Gromov-Hausdorff distance, dGH , satisfies, 1 (2) d (X, Y ) = inf (disR) GH 2 R Gromov proved that a class of metric spaces with a uniform bound, N(r, R), on the number of disjoint balls of radius r contained in a ball of radius R is GH compact [Gr]. In particular, the class of complete manifolds with Ricci ≥ −K, dim = n and diam ≤ D is precompact [Gr][Bi]. Here the PI proposes to apply and extend this theory to analyze the topological properties of manifolds with nonnegative Ricci curvature and their limits. In addition, a new space dis- tance which is weaker than the Gromov Hausdorff distance is proposed as a way of devising a precompactness theorem for manifolds with nonnegative scalar curvature. Applications of this new space distance and further generalizations will also be investigated. 1. The Topology of Manifolds with Ricci Curvature Bounded Below The topology of manifolds with Ricci curvature bounds has been an important field of study for some time. Recall that DeTurck has proven that locally the Ricci tensor determines the metric [Dt]. Lokhamp has shown that negative Ricci curvature has almost no topological restrications [Lok1][Lok2]. Thus it becomes natural to investigate manifolds with Ricci ≥ 0. Conjecture 3. [Milnor, 1969] A complete noncompact manifold with Ricci ≥ 0 has a finitely generated fundamental group. Anderson and Li have seperately proven the conjecture holds when M n has Euclidean Volume growth [And, Li]. The PI has shown it holds when the manifold has linear volume growth or small linear diameter growth [So4]. The constant in the PI’s paper has recently been improved by [XWY]. Wilking has shown that if a counter example exists then there is a counter example with an abelian fundamental group [Wlk]. 1 Other topological results concerning manifolds with nonnegative Ricci curvature have been proven as well. Yau, Itokawa-Kobayashi and Shen-Sormani have classified Hn−1(M) of such spaces [Yau][ItKo1][ItKo2] [So4][ShSo]. Schoen-Yau, Shi and others have completely classified the topology of open three dimensional manifolds with Ricci ≥ 0 [Sta] [SchYau1] [Aub] [Zh2] [AndRod][Shi]. Other important topological properties of manifolds with Ricci curvature bounds appear in [CrKr] [DaiWei] [Ich] [GlMey1][Ots1] [ShaShn] [ShnWei] [Zh1]. Examples of such man- ifolds have been constructed in [And2][And4] [AndKrLb] [Ber] [BelWei1] [BoyGal1] [BoyGalNak] [GlkPrkTus] [GlMey2] [Nab] [Ots2] [Per2] [ShaYng1] [Wlk] [Wei1] [Wr]. 1.1. Milnor Conjecture and Tangent Cones at Infinity. The PI proposes to work with Guofang Wei on the Milnor conjecture using a new approach combining our results on the properties of covering spaces and their limits with the propeties of tangent cones at infinity. Guofang Wei’s NSF grant is DMS-0204187. Wei and I have already shown that pointed Gromov Hausdorff limits of manifolds with non- negative Ricci curvature have universal covers and we have proven surjectivity results concerning the fundamental groups of the manifolds and the group of deck transforms of the limit’s univer- ¯ sal coverπ ¯1, [SoWei2]. We also proved pi1 satisfies the Theorems of Anderson, Li and Sormani mentioned above [SoWei2]. One special case of such a limit space is a tangent cone at infinity, found by rescaling a single manifold infinitely many times: Definition 4. (Y, y) is a tangent cone at infinity for (M, p) if ∃ri → ∞ such that ∀R > 0, (3) lim dGH (Bp(riR)/ri,By(R)) → 0. i→∞ Gromov proved that for any sequence ri → ∞ there is a converging subsequence, but in general the tangent cone at infinity need not be unique [Gr]. Guofang Wei and I believe we can prove the following conjecture using the work described below in Prior Results with results of Wilking [Wil]: Conjecture 5. If M n is a complete manifold with Ricci ≥ 0 that has an infinitely generated fundamental group, then it has a tangent cone at infinity with a nontrivial universal cover. It would appear then that if this tangent cone at infinity were a cone or had some sort of rescaling property, that this conjecture would lead to a contradiction and the Milnor Conjecture would be proven. However, a tangent cone at infinity is not necessarily a metric cone and need not even be polar about it’s basepoint [ChCo, Men]. Nevertheless, Guofang Wei and the PI believe that the following statement is true and can be used to prove the Milnor Conjecture. Conjecture 6. Any tangent cone at infinity of a complete manifold with Ricci ≥ 0 is its own universal cover. It should be noted, by the way, that the tangent cones at infinity for such manifolds with linear or Euclidean volume growth are their own universal covers and the same is true for the manifolds described in Anderson’s study of the fundamental group [And]. 2 Conjecture 6 is very difficult to prove using current methods to study tangent cones at infinity. Most methods used today give local properties concerning regularity, measure and topology [Ch, ChCo, SoWei2]. None give properties which are specific to tangent cones at infinity rather than other limits obtained without rescaling. The key problem is that the ri used to define a particular tangent cone at infinity can have ri+1/ri → ∞, causing the tangent cone to have no rescaling invariants at all. The PI proposes to overcome this problem by studying all tangent cones at infinity together and the relationship between them. To implement this one would start with M(n, 0), the moduli space of complete Riemannian manifolds with nonnegative Ricci curvature, and let M¯ be its closure under pointed Gromov- Hausdorff Convergence. On this space there is a rescaling flow, Φr : M¯ → M¯ . For any n M ∈ M,Φr(M) flows towards E as r → 0. The tangent cones at infinity are found by finding subsequences ri → ∞ such that Φri (M) converges. However, viewing each tangent cone at infinity seperately is like looking at a snapshot of the limit set of this flow as r → ∞. Much more information can be obtained by looking at the set of tangent cones at infinity and using the fact that this rescaling flow is defined on that set as well. Preliminary work by the PI show that the set of all tangent cones at infinity for a given manifold is closed and connected and that the recaling flow can be used to map certain tangent cones at infinity to others. We believe that by applying the results with Wei concerning covering spaces that we can prove the following two conjectures which imply Conjecture 6: Conjecture 7. For a fixed manifold with Ricci ≥ 0 if one tangent cone at infinity has a trivial universal cover, then all tangent cones at infinity have a trivial universal cover. Conjecture 8. Any manifold with nonnegative Ricci curvature has at least one tangent cone at infinity with a trivial universal cover. The PI would also like to conduct a complete investigation of the properties of the set of tangent cones at infinity. In particular the PI believes an adaption of the examples of Menguy can be used to prove the following: Conjecture 9. There is a manifold with nonnegative Ricci curvature whose set of tangent cones at infinity can be viewed as a polygon upon which the rescaling flow acts clockwise so that each side is moved along as r increases and each point is a limit as r → ∞ for one side and r → 0 for the next. Conjecture 10. There is a manifold with nonnegative Ricci curvature whose set of tangent cones at infinity form a 2 dimensional torus upon which the rescaling flow wraps densely like a curve of irrational slope on a 1x1 torus. 3 1.2. Local Loops to Infinity and Splitting. Another approach to understanding the topol- ogy of a manifold with Ricci ≥ 0 would be to use the loops to infinity property discovered by the PI in [So5]. The property states that a complete manifold with positive Ricci curvature, every noncontractible loop is homotopic to a sequence of loops diverging to infinity. It is proven using the Cheeger-Gromoll Splitting Theorem [ChGl] and was applied to control Hn−1(M, Z) in [ShSo]. Cheeger-Colding have proven an almost splitting theorem which, when combined with the Abresch-Gromoll Excess Theorem, gives very nice control of the metric’s behavior near the center of a long geodesic [ChCo][AbGl]. The PI proposes to adapt the proof of the loops to infinity property to prove the following: Conjecture 11. Given a minimizing geodesic γ :[−L, L] → M where M has Ricci ≥ −H there exists r > 0 sufficiently small that for any geodesic loop contained in Bγ(−rL)(rL/10) is homotopic to a geodesic loop contained in Bγ(rL)(rL/10).
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