Lower bounds for the volume with upper bounds for the Ricci curvature

In this note we provide several lower bounds for the volume of a geodesic ball in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.


Introduction
One of the central topics in Riemannian geometry is the relation between the curvature of a Riemannian metric defined on a manifold and the behaviour of the volume of geodesic balls. Curvature, geodesics and balls have an extremely rich relationship. A celebrated and well known result states (see for instance [2]) that if a n-dimensional Riemannian manifold (M, g) has the sectional curvatures sec M (Π) of any tangent plane Π bounded from above by a constant κ, sec M (Π) ≤ κ then, for any point p ∈ M , the volume V(p, t) of the geodesic ball of radius t centered at p is bounded from below by (1) V M (p, t) ≥ V M n κ (t) for any t ≤ min{inj(p), π/ √ κ} 1 , where V M n κ (t) is the volume of the geodesic ball of radius t in the simply-connected real space form M n κ of dimension n and constant sectional curvature κ. This inequality was obtained by Bishop and Günter and it has associated a rigidity result: if equality is attained in inequality (1), the geodesic ball of radius t in M centered at p ∈ M is isometric to the geodesic ball of radius t in M n κ . An other classical result authored by Bishop and Gromov, (see [2]) states that whenever the Ricci curvatures are bounded from below by Ric ≥ (n − 1)κ, the volume of the geodesic ball of radius t is bounded by from above by V M (p, t) ≤ V M n κ (t) for any t > 0.
Furthermore, Calabi and Yau proved (see [10]) for any complete and non-compact Riemannian manifold with Ric ≥ 0 there exists a constant C such that the volume of the geodesic ball is bounded from below by V M (p, t) ≥ C t. We would like to stress here that, in the above theorems, upper bounds are imposed only on the sectional curvature, and for the Ricci curvature only lower bounds are used. The goal of this paper is to obtain lower bounds for the volume of geodesics balls when the Ricci curvature is bounded from above. This objective is achieved in dimension 3. The results of this paper are detailed in the following section.

Main Results
Our first result is a Bishop-Günter type inequality but using bounds on the Ricci curvature: Then, for any p ∈ M and for any t ≤ min{inj(p), π/ √ κ}, the volume V M (p, t) of the geodesic ball of radius t centered at p is bounded from below by is the volume of the geodesic ball of radius t in the simply-connected real space form M 3 κ of dimension 3 and constant sectional curvature κ. The hypothesis of the above theorem implies global upper bounds in the Ricci curvature. In the following theorem we are assuming that the positive upper bound of the Ricci curvature has finite L 1 -norm in M . More precisely, for any point q ∈ M let us denote by K + : M → R the function Under the hypothesis of finite L 1 -norm of this K + function we obtain the following theorem Then, for any p ∈ M and for any t ≤ inj(p), the volume eV M (p, t) of the geodesic ball of radius t centered at p is bounded from below by An immediate consequence of the above theorem is the following corollary when we restrict ourselves to manifolds with pole Remark 2.4. Observe that in the above corollary the assumption that M contains a point with empty cut locus can not be removed. Indeed, Lohkamp proved in [7] that each manifold M n , n ≥ 3 admits a complete metric with constant negative Ricci curvature and finite volume, vol(M ) < ∞.
Outline of the paper. This paper is organized as follows: in section 3 we recall the notation and provide the propositions that will be used in section 4 to prove theorem 2.1 and 2.2. Finally in section 5 we will provide three examples where is shown that the use of lower bounds for the volume of geodesic balls obtained by using 2.1 are beter than the lower bounds obtained by using the classical Bishop-Günter inequality.

Preliminaries and notation
Given a Riemannian manifold (M, g) the curvature tensor is the (1, 3)-tensor defined by where X, Y, Z ∈ X(M ) are vector fields and ∇ is the Levi-Civita connection. Let us denote by In the space Λ 2 p M of bivectors of T p M , if {e i } is an orthonormal basis of the tangent space T p M , the inner product on Λ 2 p M is such that the bivectors {e i ∧ e j } i<j form an orthonormal basis. From the symmetry properties of the curvature tensor it is known (see [9]) that R defines a symmetric bilinear map Let v, w ∈ T p M be two linearly independent vectors, let Π be the 2-plane spanned by v and w, Π = span R {v, w} the sectional curvature of the plane Π is given by If {e 1 , · · · , e n } is an orthonormal basis of T p M , for any v, w the Ricci curvature of v, w can be obtained as for v ∈ T p M , v = 0 we can obtain an orthonormal basis { v v| , e 2 , · · · , e n } and The Ricci tensor could also be understood as the symmetric (1, 1)-tensor Moreover, the scalar curvature is given by Ric(e i , e i ) 3.1. Total Mean Curvature of Geodesic Spheres and Ricci Curvature. Let (M, g) be a n-dimensional Riemannian manifold, let p ∈ M be a point of M , and let inj(p) denote the injectivity radius of p. Let B inj(p) (0) be the ball of radius inj(p) centered at 0 in T p M , let B inj(p) (p) = exp p (B inj(p) (0)) the geodesic ball of radius inj(p) centered at p, then the exponential map is a diffeomorphism. In B inj(p) (p) \ {p} we can define polar coordinates with respect to p. Namely, for any point q ∈ B inj(p) (p) \ {p} there is associated its polar radius r(q) := dist(p, q) and its polar "angle" θ(q) ∈ S n−1 such that the unique geodesic from p to q starts at p in direction θ(q) ∈ T p M , more precisely The Riemannian metric tensor g is given with respect to the polar coordinates in B inj(p) (p) \ {p} as where (θ 1 , · · · , θ n−1 ) are coordinates in S n−1 . The radial vector field ∂r is globally defined on B inj(p) (p) \ {p} and is given by Namely, ∂r(q) is the tangent vector to the arc-length parametrized geodesic curve from p to q. Let us denote by dV g the Riemannian volume form associated to g. The volume V M (p, t) of the geodesic ball B t (p) of radius t centered at p is given by The vector field ∂r coincides with the gradient ∇r of the polar radius function r on B inj(p) (p) \ {p}, i.e., ∂r = ∇r, and furthermore ∇r = ∂r = 1. Moreover since the geodesic sphere S t (p) of radius t centered at p is a level set of r, i.e., S t (p) = r −1 (t) then the vector field ∇r is a unit vector field normal and pointed outward to S t (p).
The volume A M (p, t) of the geodesic sphere S t (p) of radius t centered at p, is given therefore by where ∇r dV g is the contraction of the Riemannian volume form dV g with the vector field ∇r. In order to simplify the notation we will make use of is smooth and with derivative A M (p, t). The second fundamental form α of the inclusion map from S t (p) to M is given in terms of the Hessian Hess M r of the geodesic distance funtion r to the pole p because for any two vector fields X, Y ∈ X(S t (p)) The mean curvature vector field H of S t (p) is given therefore in terms of the Laplacian ∆ M r of the distance function to p because for any q ∈ S t (p) and any orthonorl- Hess M r(E i , E i )∇r = −∆ M r∇r and the scalar mean curvature pointed inward of S t (p) at q ∈ S t (p) is thus The following proposition states the first and second variation formula for the area function t → A M (t), Proposition 3.1. Let (M, g) be a Riemannian manifold, suppose that p ∈ M and t < inj(p). Then, (1) The first derivative A ′ M (p, t) with respect to t of the volume A M (p, t) of the geodesic sphere S t (p) of radius t centered at p is given by HdA g .
(2) The second derivative A ′′ M (p, t) with respect to t of the volume A M (p, t) of the geodesic sphere S t (p) of radius t centered at p is given by Proof. For any q ∈ B inj(p) (p) \ {p} the vector field ∇r induces a 1-parameter group of diffeomorphisms φ s : B inj(p) (p) \ {p} −→ B inj(p) (p) \ {p} given by (see [5] for instance) φ 0 (q) = q d ds φ s (q) s=0 = ∇r(q). hence d ds r(φ s (q)) = ∇r, ∇r = 1 Therefore r(φ s (q)) = r(q) + s. Namely, φ h is a diffeomorphism Using this diffeomorphism we can compute the first and second variation of the volume A M (p, t). For the first derivative where here φ * h ∇r dV g is the pullback of ∇r dV g by the diffeomorfism φ h . Let us denote by

But observe that lim
h→0 ω h = L ∇r ∇r dV g here L ∇r ∇r dV g denotes the Lie derivative of ∇r dV g along ∇r. Now we are proving that we can introduce the limit into the integral in (4). In order to prove that we are using a parametrization of S t (p). Let D 1 , · · · , D k be open domains of integration in R n−1 and let F i : D i → S t (p) be smooth maps such that (1) F i restricts to an orientation-preserving diffeomorphism from D i onto an open subset W i ⊂ S t (p); Then, (see proposition 16.8 of [6]), Let us integrate using the coordinates (x 1 , x 2 , · · · , x n−1 ) of R n−1 where ω i h is the smooth function given by h is also continuous in h. Moreover, since S t (p) is compact, for any t 0 ≥ 0 and any i ∈ {1, · · · , k} there exists C such that By using the Dominated Convergence Theorem (see [1]), But by the Cartan's magic formula (see [6]) L ∇r ∇r dV g = d (∇r ∇r dV g ) + ∇r d (∇r dV g ) = ∇r d (∇r dV g ) = ∇r div(∇r)dV g = div(∇r)∇r dV g = ∆ M rdA g , For the second derivative, likewise, Let us choose now an orthonormal basis {E 1 , · · · , E n−1 , ∇r} of T q M which diagonalizes Hess M r, i.e., Taking into account that α(E i , E j ) = −Hess M r(E i , E j )∇r and using the Gauss formula (see [8] for instance) Finally, it is easy to check that When the dimension of M is 3, the geodesic sphere S t (p) of radius t centered at p has dimension 2 and the scalar curvature is given in terms of the Gaussian curvature K G , scal St(p) = 2K G and by the Gauss-bonnet Theorem therefore we can state the following corollary to proposition 3.1 Corollary 3.2. Let (M, g) be a 3-dimensional Riemannian manifold, let p ∈ M be a point of M . Then for any 0 < t < inj(p), the second derivative A ′′ M (p, t) with respect to t of the volume A M (p, t) of the geodesic sphere S t (p) of radius t centered at p is given by where scal M is the scalar curvature function of M , and Ric(∇r, ∇r) is the Ricci tensor evaluated in ∇r.
A ′′ M (p, s)ds Therefore using the above corollary, Ric(∇r, ∇r)) dA g ds Taking the limit t 0 → 0 we obtain the following Proposition 3.3. Let (M, g) be a 3-dimensional Riemannian manifold, let p ∈ M be a point of M . Then for any 0 < t < inj(p) Proof. The proposition follows taking the limit t 0 → 0 in equation (5) Indeed, in [3] for example, it is proved that  Then, for any p ∈ M and for any t ≤ min{inj(p), π/ √ κ}, the volume V M (p, t) of the geodesic ball of radius t centered at p is bounded from below by  (7) can be rewritten as

Proof of Theorem 2.2. The statement and proof of theorem 2.2 is as follows
Theorem. Let (M, g) be a 3-dimensional Riemannian manifold. Suppose that Then, for any p ∈ M and for any t ≤ inj(p), the volume eV M (p, t) of the geodesic ball of radius t centered at p is bounded from below by Proof. Let {∇r, E 1 , E 2 } be an orthonormal basis of T q M . Since Ric ≤ K + , By using proposition 3.3 and the theorem follows integrating twice and taking into account that V M (p, 0) = V ′ M (p, 0) = 0, V M (p, t) ≥ 4 3 πt 3 − Ct 2 .