A New Proof of the Cheeger-gromoll Soul Conjecture and the Takeuchi Theorem

Let Mn be a complete, non-compact Riemannian manifold with nonnegative sectional curvature. We derive a new broken flat strip theorem associated with the Cheeger-Gromoll convex exhaustion, in the case when Mn is not diffeomorphic to Rn. This leads to a new proof of the Cheeger-Gromoll soul conjecture without using Perelman’s flat strip theorem. Using the Cheeger-Gromoll inward equidistant evolution and the Calabi’s barrier surface technique, we also provide a new proof of the Takeuchi Theorem. §1. A new broken flat strip theorem and applications to the soul conjecture Suppose that M is a C∞-smooth, complete and non-compact Riemannian manifold with nonnegative sectional curvature. Cheeger-Gromoll [ChG] established a fundamental theory for such a manifold. Among other things, they showed that M admits a totally convex exhaustion {Ωu}u≥0 of M, where Ω0 = S is a totally geodesic and compact submanifold without boundary. Furthermore, M is diffeomorphic to the normal vector bundle of the soul S. In particular, if the soul S is a point, then M is diffeomorphic to the Euclidean space R. The Cheeger-Gromoll soul conjecture asserts that “if a complete and non-compact Riemannian manifold M has nonnegative sectional curvature and if M contains a point p0 where all sectional curvatures are positive, then M must be diffeomorphic to the Euclidean space R”. This is true if M has positive sectional curvature everywhere by the earlier work of Gromoll and Meyer [GrM]. This conjecture was solved by G. Perelman [Per] by his remarkable flat strip theorem. Earlier partial results on the Cheeger-Gromoll soul conjecture were obtained by Marenich, Walschap and Strake, see references in [Per]. If there is a totally geodesic isometric immersion φ : R× [0, `] → M (s, t) → φ(s, t), (1.1) Both authors are partially supported by NSF Grants. The first author is grateful to MSRI and Max-Planck Institute for Mathematics at Leipzig for their hospitality. Typeset by AMS-TEX 1 then φ ( R× [0, `]) is said to be an immersed flat strip in M. If {Ωu} is a CheegerGromoll convex exhaustion, it is known that, for each u > 0, Ωu is a totally convex (and hence totally geodesic) submanifold with boundary. We let ∂Ωu be the relative boundary of Ωu. A flat strip φ ( R× [0, `]) is said to be compatible with the CheegerGromoll convex exhaustion {Ωu} if each horizontal geodesic φ ( R× {t}) of the flat strip is contained in ∂Ωu(t) for some u(t) > 0 where t > 0. In this paper, we obtain a different type of flat strip theorem using a different approach. Theorem 1.1. Let M be a complete and non-compact Riemannian manifold with nonnegative sectional curvature. Suppose that M is not diffeomorphic to the Euclidean space R. Then for each x ∈ M there is a non-trivial broken flat strip Φ = {φi}1≤i≤N passing through x. Moreover, each flat strip φi is compatible with the Cheeger-Gromoll exhaustion. Consequently, then the Cheeger-Gromoll soul conjecture is true. A result similar to Theorem 1.1 was obtained in [CaS] via a different method. The broken flat strip stated in Theorem 1.1 above is indeed determined by the Cheeger-Gromoll exhaustion {Ωu} in a canonical way, which we now describe in the next subsection a. The Cheeger-Gromoll broken geodesics. Let us briefly recall the Cheeger-Gromoll convex exhaustion. Definition 1.2. (A Cheeger-Gromoll exhaustion) For a complete and non-compact Riemannian manifold M. A Cheeger-Gromoll exhaustion {Ωu}u≥0 has the following properties. There is a partition a0 = 0 < a1 < ...am < am+1 = ∞ of [0,∞) and an exhaustion {Ωu}u≥0 of M such that the following holds: (1.2.1) M = ∪u≥0Ωu. If u > am then dim[Ωu] = n. If u ≤ am, then dim[Ωu] < n. (1.2.2) Ω0 = S is the soul of M, which is a totally geodesic C∞-smooth compact submanifold without boundary. (1.2.3) If u > 0, Ωu is a totally convex, compact subset of M and hence Ωu is a compact submanifold with a C∞-smooth relative interior. Furthermore, dim(Ωu) = ku > 0 and Ωu has a non-empty (ku − 1)-dimensional relative boundary ∂Ωu; (1.2.4) For any u0 ∈ [aj , aj+1] and 0 ≤ u ≤ u0 − aj , the family {Ωu0−u}u∈[0,u0−aj ] is given by the inward equidistant evolution: Ωu0−u = {x ∈ Ωu0 |d(x, ∂Ωu0) ≥ u}. (1.2) (1.2.5) If u > am then u − am = max{d(x, ∂Ωu)|x ∈ Ωu)}. If 0 ≤ j ≤ m − 1 then aj+1 − aj = max{d(x, ∂Ωaj+1)|x ∈ Ωaj+1} and hence dim[Ωaj ] < dim[Ωaj+1 ] for j ≥ 0. 2 For each compact convex subset Ω ⊂ M, we let U2(Ω) = {x ∈ Mn|d(x, Ω) < 2}. Its cut-radius is given by δΩ = sup{2| there is a unique nearest point projection PΩ : U2(Ω) → Ω}. For each given T > am, there is an estimate for the cut-radius of convex subsets A in ΩT . For each x ∈ M, we let InjMn(x) be the injectivity radius of M at x. Similarly, let InjMn(A) = sup{InjMn(x)|x ∈ A}. Lemma 1.3. ([ChG, p425]) Let A ⊂ ΩT be a connected, convex and compact subset in a Riemannian manifold M with nonnegative sectional curvature and let K0 = max{K(x)|x ∈ ΩT+1}, InjMn(ΩT ) be the upper bound of sectional curvature on ΩT+1 and S be as above. Suppose that dim(ΩT ) = n. Then the subset A has cut-radius bounded below by δA ≥ δ0(T ) = 14 min{InjMn(ΩT ), π √ K0 , 1}, where δ0(T ) is independent of choices of A with A ⊂ ΩT . Proof. Suppose there are two points q1 6= q2 in A and p ∈ Uδ0(T )(A) such that d(p, q1) = d(p, q2) = d(p,A) = `. Let σi the geodesic segment from qi to p. By the first variation formula, the angle between σ′ i(0) and the vector tangent to A is at least π2 . Comparing the inner angles of geodesic triangles of the same side lengths in the round sphere of curvature K0, one obtain the angles 0 < ∠q1(p, q1) < π2 , a contradiction. ¤ For given T > am, we choose a partition u0 = 0 < u1 < ... < uN = T such that {ai}0≤i≤m is a subset of {uj}; and Ωuj ⊂ Uδ0(T )(Ωuj−1). Definition 1.4. (The Cheeger-Gromoll broken geodesic) Let {Ωu} be a CheegerGromoll convex exhaustion. For each T > am, let δ0(T ) and the partition {uj} be as above. If Pj−1 : Ωuj → Ωuj−1 is the nearest point projection, then for each x ∈ ΩT , we let xN = x, xN−1 = PN−1(x), ..., xj−1 = Pj−1(xj) for j = N, N − 1, ..., 1. The broken geodesic σ = {σj} joining x0, x1, ..., xN = x is called a Cheeger-Gromoll broken geodesic from x0 ∈ S to xN = x. In Definition 1.4 above, the points {xj} are not necessarily distinct. b. The monotone property of tangent cones of the Cheeger-Gromoll exhaustion Suppose that σ = {σj} is a Cheeger-Gromoll broken geodesic from the soul to x as above. Assume xi−1 6= xi for some i. We consider the geodesic segment σi : [0, `i] → M from xi−1 to xi. If xi ∈ ∂Ωwi for some wi > 0, we let ui(t) = wi − d(σi(t), ∂Ωwi). In this case, we have σi(t) ∈ ∂Ωui(t). 3 In what follows, we let T− y (Ω) be the tangent cone of Ω at y: T− y (Ω) = {~v ∈ Ty(M)| lim sup t→0+ d(Expy(t~v), Ω) t = 0}. The following monotone theorem plays an important role in the proof of Theorem 1.1. Theorem 1.5. Let Pσ be the parallel transport along a Cheeger-Gromoll broken geodesic σ = {σj} and ui(t) be as above. Then the tangential cones of {Ωu} are monotone under the parallel transport along a Cheeger-Gromoll broken geodesic: Pσ [ T− σi(t0)(Ωui(t0) ] ⊂ T− σi(t1)(Ωui(t1)) for any pair 0 ≤ t0 ≤ t1 ≤ `i. For the proof of Theorem 1.5, it is sufficient to consider the normal cones of Ωu instead. Definition 1.6. (1) Let Ω be a convex subset of M and σ : [0, `] → M be a geodesic with σ(0) ∈ Ω. The geodesic σ is called at least normal to Ω if the angle between σ′(0) and any vector tangent to Ω is at least π2 . (2) The normal cone of Ω in M is defined by N+(Ω,Mn) ={(p,~v)|p ∈ Ω, d(Expp(t~v),Ω) = t|~v|, for 0 ≤ t|~v| < δΩ}. Similarly, one can define the relative normal cone N(Ωu, int(Ωu+2)) (3) (Minimal normal vector) Let Ωu, Ωu+2 and N(Ωu, int(Ωu+2)) be as above. Let σ(p,~v) : [0, 2] → M be a geodesic given by σ(p,~v)(t) = Expp(t ~v |~v| ), where ~v 6= 0. If σ(p,~v) is a length-minimizing geodesic from p ∈ Ωu to ∂Ωu+2, then ~v is called a minimal normal vector in N+ p (Ωu, int(Ωu+2)). Proof of Theorem 1.5. We need to choose 2 sufficiently small so that (1) there is a nearest point projection P : int(Ωu+2) → Ωu; and (2) Ωu = {x ∈ Ωu+2|d(x, ∂Ωu+2) ≥ 2} holds. We first find j so that aj ≤ u < aj+1 for some 0 ≤ j ≤ m. Let T = u + am + 1 and δ0(T ) be given by Lemma 1.3. It follows from a result of Yim that there is a constant CT such that, for 0 ≤ a < b ≤ T , we have max{d(x, Ωa)|x ∈ Ωb} ≤ CT (b− a), (1.3) see [Y2, Theorem A.5(3)]. In what follows, we always choose 0 < 2 = 2u < min{[aj+1 − u], δ0(T ) 2CT }. (1.4)