A direct proof of one Gromovs theorem

arXiv:0802.0098v1 [math.DG] 1 Feb 2008 A DIRECT PROOF OF ONE GROMOV’S THEOREM YU. D. BURAGO, S. G. MALEV, D. I. NOVIKOV‡ Abstract. We give a new proof of the Gromov theorem: For any C > 0 and integer n > 1 there exists a function ∆C,n such that if the Gromov– Hausdorffdistance between complete Riemannian n-manifolds V and W is not greater than δ, absolute values of their sectional curvatures |Kσ| ≤C, and their injectivity radii ≥1/C, then the Lipschitz distance between V and W is less than ∆C,n(δ) and ∆C,n →0 as δ →0. 1. Introduction Denote by M(ρ, C, n) the class of complete n-dimensional Riemannian manifolds V with section curvatures |Kσ| ≤C < ∞and the injective radii rin(V ) ≥ρ, where C, ρ are some positive constants. There are two well-known metrics on this class: the Lipschitz metric and the Gromov–Hausdorffmetric. Recall that the Lipschitz distance dLip(X, Y ) between metric spaces X, Y is defined as dLip(V, W) = ln inf{k : Bilipk(V, W) ̸= ∅}, where Bilipk(V, W) denotes the class of all bi-Lipschitz homeomorphisms between V and W with bi-Lipschitz constant k ≥1. By bi-Lipschitz constant of a homeo- morphism ζ we mean maximum of Lipschitz constants for ζ and ζ−1. Instead of the Gromov–Hausdorffmetric, we use a metric, equivalent to it (see, for instance, [1]). We preserve notation dGH for this metric. By definition, the distance dGH(V, W) is the infimum of all δ > 0 with the property that there exists a mapping χ : V →W such that χ(V ) is a δ-net in W and χ changes distances by at most on δ: |dW (χ(x), χ(y)) −dV (x, y)| < δ for any points x, y ∈V . Note that χ is not supposed to be continuous. The purpose of this paper is to give a direct proof for the following Gromov theorem. Theorem 1 (Gromov [4], page 379). For given ρ > 0, C > 0 and an integer n > 1, there exists a positive function ∆= ∆(C,n,ρ) such that ∆(δ) →0 as δ →+0 and if V, W ∈M(ρ, C, n) satisfy the condition dGH(V, W) < δ then dLip(V, W) < ∆(δ). In contrast to Gromov’s proof using axillary embeddings of the manifolds V , W into an Euclidean space of a large dimension, we directly construct a bi-Lipschitz diffeomorphism h(x) between V and W with a required bi-Lipschitz constant ∆(δ). The map h(x) is obtained by gluing together “local maps” ϕi with help of partition of unity. The maps ϕi are defined on some balls B2ε(vi) ⊂V which form a locally The first author is partially supported by RFBR grant 05-01-00939. The third author is supported by Samuel M. Soref & Helene K. Soref Foundation. 1 2 YU. D. BURAGO, S. G. MALEV, D. I. NOVIKOV‡ finite covering of V . This gluing is based on Karcher’s center of mass technique [6]. The resulting map turns out to be bi-Lipschitz with the required constant since the mappings ϕi are C1-close one to another on the intersection of their domains. To justify publishing our proof, note that though ideas of Gromov’s proof ex- plained very clear in his book, some details are omitted in his exposition. Later on C denotes different constants depending on n = dim V = dim W only. We always assume δ to be sufficiently small, δ < δ0, where δ0 depends on n only. All these constants can be computed explicitly, if such a need arises. 2. Preparations By a suitable rescaling of V, W, one can get rid of one parameter and assume that the absolute values of section curvatures smaller than δ and the injectivity radii are bigger than δ−1. Also we can assume that and dGH < δ. We always suppose that 0 < δ ≪1 and denote ε2 = δ ≪1. (1) 2.1. ε-Orthonormal base. We say that a basis {e1, . . . , en} ⊂Rn is ε-orthonormal if |(ei, ej) −δij| < ε for all i, j = 1, . . . , n, where δij is the Kronecker symbol. A linear map L: M →N of two Euclidean spaces is ε-close to isometry if ∥L−Q∥< ε for some isometry Q: M →N. Lemma 1. Let {ξi} be an ε-orthonormal base of Rn, ε < 1 2n. Let L : Rn →Rn be a linear operator. If ∥L(ξi)∥< δ, i = 1, . . . , n then ∥L∥< 2√nδ. Assume that 8n√nδ < 1 and |⟨Lξi, Lξj⟩−⟨ξi, ξj⟩| < δ, i, j = 1, .., n. Then L is 8n√nδ-close to an isometry. Proof. Lemma could be easily proved by straightforward calculation. We give a proof to the first part, the second part can be obtained the same way. Let x = P xiξi be a unit vector. Then 1 = ⟨x, x⟩= X i,j xixj⟨ξi, ξj⟩≥ n X i=1 x2 i −ε n X i=1 |xi| !2 ≥(1 −nε) n X i=1 x2 i . Therefore P x2 i ≤(1 −εn)−1 ≤2. Thus ∥Lx∥= ∥ n X i=1 xiL(ξi)∥≤δ n X i=1 |xi| ≤δ q n X x2 i ≤2δ√n. □ 2.2. On comparison theorems and exponential mappings. Denote by J a Jacobi vector field along a geodesic γ : [0, 2] →V . As usually, ˙J(t) = D dtJ means the covariant derivative of J along γ. We need a known comparison theorem of Rauch-style, see [3], 7.4, or the original Karcher’s paper [6]. We formulate the theorem in the form adopted to our case. A DIRECT PROOF OF ONE GROMOV’S THEOREM 3 Theorem 2. Suppose that all section curvatures |K| ≤δ and 0 < δ ≪1 (it is enough to have δ < 10−2). Then |J(0)| cos( √ δt) + | ˙J(0)|δ−1/2 sin( √ δt) ≤|J(t)| ≤|J(0)| cosh( √ δt) + | ˙J(0)|δ−1/2 sinh( √ δt). (2) As usually, we apply this esti

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