An effective estimate on Betti numbers

We provide an effective estimate on the Betti numbers of the loop space of a compact manifold which admits a finite Grauert tube. It implies the polynomial estimate in \cite{Chen} after taking the radius of the tube to infinity.


Introduction
For a compact Riemannian manifold (M, g), we say that it admits a Grauert tube of radius R if there exists a canonical complex structure on the disc bundle of radius R. It is denoted by T R M. The R is called the radius. Such a complex structure which is called the adapted complex structure (cf. Theorem A of [13]) is uniquely determined by the Riemannian structure of (M, g) and makes M a totally real embedded submanifold. The concept arises in the study of the homogenous complex Monge-Ampère equation [2,12] and the complexification of a real analytic manifold [5]. Please refer to [9] and reference therein for more detailed descriptions of its development. Its existence is not automatic. In fact it was proved by Lempert (cf. Theorem 1.5 of [8]) that the existence of a Grauert tube implies that g must be a real analytic metric. Hence it does put some constraint on the metric g besides requiring that M is real analytic. Conversely for a compact real analytic manifold M and a real analytic metric there always exists a real R 0 > 0 such that T R 0 M is a Grauert tube. Namely there exists a canonical adapted complex structure on the disc bundle T R 0 M. The precise definition of the adapted complex structure requires introducing additional notions which we defer to the next section. The main result of this note is Theorem 1.1 Let (M, g) be a simply connected compact Riemannian manifold which admits a Grauert tube of radius R. Given any coefficient field F, and a positive integer k, then there exists a constant C > 0, which is independent of F and k, such that Here (M) denotes the space of continuous loops, ω k is the volume of the kdimensional unit sphere S k .
Taking R → ∞, one can recover the polynomial (in terms of k) estimate of Chen [3] in the case when M admits a global Grauert tube.

Corollary 1.2 Let (M, g) be a compact Riemannian manifold which admits a global Grauert tube. Then
It follows that M is rational elliptic, by the result of Félix and Halperin (cf. page 110 of [10]), and Proposition 5.6 of [10]. This provides a positive answer to a special case of a conjecture of Bott (which asserts that any simply connected close manifold with nonnegative sectional curvature must be rationally elliptic, in view of the result of Lempert-Szöke below, since the existence of a global Grauert tube implies the nonnegativity of the sectional curvature). Theorem 1.3 (Lempert-Szöke) Let (M, g) be a simply connected compact Riemannian manifold which admits a Grauert tube of radius R. Then the sectional curvature of (M, g) is bounded from below by − π 2 4R 2 . In particular, if R = ∞, (M, g) has nonnegative sectional curvature.
For a fixed point p ∈ M let D( p, R) denote the set of vectors in the tangent space T p M satisfying |v| < R. Namely D( p, R) = {v ∈ T p M | |v| < R. Let n( p, R, x) be the number of the pre-images of x ∈ M in the tangent space T p M under the exponential map at p which are inside D( p, R). Namely n( p, R, x) = #{v ∈ T p M | |v| < R, exp p (v) = x}. A result of Gromov, via the Morse theory on the energy functional defined on the space of pathes, asserts the following useful estimate on a compact simply connected manifold M (cf. [10], Theorem 5.10, formula (5.3) and the estimate above it on page 124, and Remark 5.28) for a regular value x of the exponential map exp p : Note that the right hand side is finite away from a measure zero set. Here C is a constant independent of F and k. Hence On the other hand, the area formula for Lipschitz maps (cf. [4], provided that T ≥ Diam(M). Here (σ, θ ) is the polar coordinate of T p M, and S n−1 is the unit sphere in T p M. It is well known that Jac(exp p )| (σ,θ) can be computed via the square root of the determinant of the gram matrix of the n − 1 normal Jacobi fields Here after adding e n =γ (0) = θ (assuming that γ is parametrized by the arc-length), {e i } 1≤i≤n forms an orthonormal frame of T p M. We denote also their parallel transport as {e i }. Combining them we have Note that if we let J n (σ ) = σ e n (σ ), the right hand side remains the same if we replace the (n−1)×(n−1) matrix with the n×n matrix ( J i , J j ) 1≤i, j≤n and then compensate with a factor of 1 σ since J i , J n = 0 for 1 ≤ j ≤ n − 1 and |J n | 2 = σ 2 . Note that the volume comparison fails beyond the first conjugate locus, hence could not be applied directly otherwise the conjecture of Bott would have been known many years ago. Here we show that it can be estimated under the additional structure of the Grauert tube via the analytic continuation. One would naturally conjecture that Theorem 1.1 (or the estimate on det( J i (σ ), J j (σ ) ) 1≤i, j≤n−1 ) holds for a closed manifold with a sectional curvature lower bound − π 2 4R 2 , but without any assumption on the existence of the Grauert tube (in particular no real analyticity assumption on the metric).

Proof of the Theorem
The proof utilizes the framework developed, and results obtained, in the important paper of Lempert and Szöke [9]. Below we need some results from that paper after some basics on geodesic flows on which one can find more detailed coverage in some excellent books e.g. [7,10,11].
The idea is to use the existence of a holomorphically immersed strip S R = {z = σ + √ −1τ ∈ C | 0 < τ < R}, whose closure passes p, to estimate the right hand side of (1.5) via a Fatou Lemma (for positive harmonic functions) by adapting the considerations of [9] to prove the estimate (1.1). We include some introductory materials (mostly from [7,9,11]) for the convenience of the readers.
We also abbreviate Dτ M as (τ M ) * . The kernel of (τ M ) * is called the vertical subspace of T u T M (denoted by V u ). We may identify the vertical tangent at u, given locally as ( The connection map K : dt denotes the covariant derivative with respect to the Levi-Civita connection. Namely it is the local derivative of the parallel vector extension along the direction given by X . Clearly the definition does not depend on the choice of c(t).
dt | t=0 (see also the proposition below). The kernel of K at u is called the horizontal subspace H u . Clearly V u ∩ H u = {0} and T u T M = V u ⊕ H u by the dimension counting. The following (cf. Proposition 4.1 of Ch II [11]) is well known.

Proposition 2.1 For y ∈ T p M and X a smooth vector field (which is viewed as a map M → T M), K (D X(y)) = ∇ y X.
From the above discussion it is clear that the map j u : In terms of the local coordinate expression above, For any smooth map f : M 1 → M 2 between two manifolds, let f * : T M 1 → T M 2 be the associated differential map which sends (x, Abusing the notation we have that γ * (σ 1 + √ −1 · 0) = γ (σ 1 ), namely we write (γ (σ 1 ), 0) simply as γ (σ 1 ). As γ runs among all geodesics, γ * : which is called the Riemann foliation. The uniqueness of the geodesic with a given initial point and a velocity vector implies that it is indeed a foliation.
The adapted holomorphic structure can be defined as follows (cf. [13]). (M, g) be a complete Riemannian manifold. For given R, a smooth complex structure on the manifold T R M will be called an adapted holomorphic structure if for any geodesic γ : R → M, the map γ * : γ −1 * (T R M) → T R M is holomorphic. When such adapted holomorphic structure exists the disc bundle T R M is called the Grauert tube of radius R.

Definition 2.2 Let
Given the adopted holomorphic structure, an important object is the so-called parallel vector field (perhaps a more suitable name is the generalized Jacobi field) associated with γ * . It is defined as the variational vector field of (γ t ) * for a family of geodesics is not hard to see that there exists a generalized Jacobi field (parallel vector field) ξ such that ξ( √ −1τ 1 ) = η. (Since γ is not parametrized by the arc length it is sufficient to consider The parallel vector field generalizes the concept of the Jacobi field, since by the way it is defined in the last equation we omit the second component since it is zero), which is the variational vector field of a family of geodesics, hence a Jacobi field. Namely J (σ ) = ξ | R (σ ) is a Jacobi field. The Jacobi field is defined along the image of a geodesic γ , the generalized Jacobi field (parallel vector field) is defined along the image of γ * , namely a leave of the Riemann foliation.
In the above if we choose τ 1 = 1, τ M (φ σ (z(t))) is a family of geodesic γ t (σ ). Hence 3 of Ch 2 of [11]). The discussion works similarly for any τ 1 > 0. The holomorphicity of (γ t ) * for a family of geodesics implies that the holomorphic component (namely the (1, 0)-part) of the variational vector field, the generalized Jacobi field (parallel vector field), is holomorphic by calculations with respect to the holomorphic coordinates (cf. Proposition 5.1 of [9]). Now for an orthonormal frame {v j } n j=1 as the above we choose ξ j and η j at u ∈ T M such that dτ M (ξ j ) = v j and K (ξ j ) = 0 and dτ M (η j ) = 0, K (η j ) = v j and then extend them as above into 2n parallel vector fields along a Riemann foliation γ * (σ + √ −1τ ). By Lemma 5.1 of [9], the holomorphic parts, ξ 1,0 j and η 1,0 j are holomorphic over the domain where γ * is holomorphic. In the case that γ (σ ) is parametrized by the arc-length, they are holomorphic on the strip S R = {σ + √ −1τ | 0 < τ < R} if T R M is a Grauert tube with the adapted holomorphic structure. Here we may choose τ 1 small such that u ∈ T R p M in the construction given in the previous two paragraphs. The following proposition summarizes the main construction of [9]. L. Ni Proposition 2.3 Let = (ξ 1 , · · · , ξ n ) and H = (η 1 , · · · , η n ) and 1,0 and H 1,0 be the holomorphic components. Then (i) The 2n-vectors {ξ j , J (ξ j )} n j=1 are linearly independent on the strip S R and {ξ 1,0 j } are linearly independent over C on the strip (here J is the almost complex structure of the tube and ξ 1,0 The 2n-vectors η 1 , · · · , η n , ξ 1 , · · · , ξ n are linearly independent; Their restrictions to R are Jacobi fields J 1 (σ ), · · · , J n (σ ), J n+1 (σ ), · · · , J 2n (σ ) satisfying that for 1 ≤ j ≤ n, There exist a holomorphic matrix f on the strip S R , which, after being extended to R, may have poles on a discrete subset of R, such that

Proof
The results were mainly proved in Sect. 6 of [9] by exploiting the Kähler/symplectic structure on T R M. By the way in which ξ j and η j are constructed it is clear that they are linearly independent at u ∈ T M due to that j u is an isomorphism. Then their extensions are linearly independent (over R) due to the property of the geodesic flow φ σ , in particular, it is an isomorphism between tangent spaces of the domain and target points (cf. Propositions 1.90 and 1.92 of [1]). The linear independence of {ξ 1,0 } over C needs to use the Kähler form on T R M, which was proved in Proposition 6.4 of [9]. The positivity of Im f is proved in Lemma 6.7 of [9].
The holomorphicity of f near 0 is due to the fact that {ξ j (σ )} are linearly independent for σ small. The equations satisfied by f (0) and f (0) are easy consequences of the constructions of {ξ j } and {η j }.
Applying the arguments/proofs in [9] we also have the following result.

Proposition 2.4 Under the assumption that M admits a Grauert tube of radius R we have the following results.
(i) The 2n-vectors {η j , J (η j )} n j=1 are linearly independent on the strip S R and {η 1,0 j } are linearly independent over C on the strip; (ii) There exist a holomorphic matrixf on the strip S R , which, after being extended to R, may have poles on a discrete subset of R, such that 1,0 (z) = H 1,0 (z)f (z) with z = σ + √ −1τ ∈ S R ;f = f −1 , hence symmetric over the points wheref is finite; (iii) Im(f ) is symmetric and negative definite for σ + √ −1τ with τ > 0; On R,f is real and = Hf , provided thatf is finite; Andf has a pole at z = 0 and is finite for σ = 0 small. Moreoverf extends to R \ S with S being the set {σ j } ∪ {0} where {γ (σ j )} is the set of the conjugate points with respect to γ (0) = p.
Proof First by Proposition 6.6 and the proof of Proposition 6.4 in [9] we have that {η 1,0 j (z)} are linearly independent for τ > 0. Hence there exists a matrix valued holomorphic functionf such that 1,0 (z) = H 1,0 (z)f (z). Namely f (z) is invertible withf = f −1 for τ > 0. Hencef and Imf are symmetric. This proves (i) and (ii). By the exactly same argument as that of Proposition 6.8 of [9], Imf is invertible for τ > 0. Now we consider the expansion off near 0: This implies that Im(−f ) is positive definite at z = √ −1τ with τ small. The analyticity off and the fact that Imf is invertible for τ > 0 imply that Im(−f ) is positive definite for z ∈ S R . This proves the first part of (iii).
Since on R, f is real and = Hf over the points wheref is finite. Hencef does not have a pole, except possibly at σ j , where γ (σ j ) is conjugate to p along γ from the construction of ξ j (σ ) and η j (σ ). Since {η j | R (σ )} are zero at σ = 0, and are linearly independent for σ small, and being linearly independent for σ if γ (σ ) is not conjugate to p along γ , the last part of (iii) holds.
Note that both f (z) and −f (z) are valued in the Siegel upper-half space of degree n. The following result provides the key ingredient for estimating the Jacobian of the exponential map. Here we identify the (σ ) and H(σ ) with their matrices representation, with respect to a orthonormal frame {e i } obtained by parallel transplanting {e i }, a frame at p, along γ (σ ).
Proof First observe that for a holomorphic F(z) with z = σ + √ −1τ defined on S R , if F = U + √ −1V , with U being the real part and V being the imaginary part, the Cauchy-Riemann equation implies that F z = U σ + √ −1V σ . In the case that F| R is real we have that F z = U σ = F σ . Namely by abusing the notation we denote by F both the complex derivative and the d dσ when F has a finite extension near a point σ 0 ∈ R. With the above consideration σ = 2 ( 1,0 z ) = 2 (H 1,0 zf +H 1,0f z ) on S R and over the domain wheref has a holomorphic extension. Restrict to R \ S, wheref is finite, we have that the right hand side equals to H σf + Hf σ sincef is real valued there. Namely σ = H σf +Hf σ holds on R\ S, which we abbreviate as = H f +Hf . The second identity follows from the first by plugging ( tr ) = (f tr ) H tr +f tr (H tr ) into the first equation, and noting that H tr H −(H tr ) H = 0 by Proposition 6.10 of [9] (the argument below also provides a simple proof).
The first identity follows from that (i) d dσ ( tr H −( tr ) H) = 0, which is a consequence of the Jacobi equation; and (ii) tr H −( tr ) H (0) = id . This argument also provides a simple proof of Proposition 6.10 of [9].
The last identity was proved in the proof of Proposition 6.11 of [9] for σ small. To see that it holds on R \ S with S being the set of the poles of f we can apply the same argument as above. Namely plug (H) = f + f , which holds on R \ S , into the first identity, and note that tr − ( tr ) = 0 on R, we have the last identity for all σ ∈ R \ S . By a calculation similar to the one in the proof of Proposition 7.1 of [9], the Jacobi curvature R γ (v) = R(v,γ )γ ) has the expression in terms of the Schwarz derivative off over the σ ∈ R \ S, with S being the set of the pole off (which contains {σ j } with {γ (σ j )} being the conjugate points of γ (0) = p along γ ): The result can also be derived from the invariance of the Schwarz derivative under the projective transformation and Proposition 7.1 of [9]. Namely if we denote the Schwarz Recall that for any nonsingular family of

Corollary 2.6
Away from a discrete subset S ⊂ R, In particular .
(2.5) Theorem 1.1 then follows by a simple calculation.
This, together with (2.4) and the positivity of the measure μ, implies that Taking determinant on the both sides and noting t = e πσ R , (2.5) follows from Proposition 2.5 and calculations.
Finally to get the estimate in Theorem 1.1 one simply observes that when restricted to R, f jn = 0, for 1 ≤ j < n, and f nn = σ , and applies the above argument to the up-left (n − 1) × (n − 1) sub-matrix of f .