A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound and Its Applications

Copyright q 2010 Zisheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.


Introduction
In comparison geometry of Ricci curvature, the classical Bishop-Gromov volume comparison has many applications, such as at least the linear volume growth of complete noncompact Riemannian manifolds with nonnegative Ricci curvature see 1 , the upper bound of total Betti number growth of Riemannian manifolds see 2-4 , and the finite topological type of complete noncompact Riemannian manifolds with nonnegative Ricci curvature or quadratic Ricci curvature decay see 3, 5, 6 .
In 7 , Lott and Shen establish a volume comparison estimate with quadratic Ricci curvature decay, and apply it to investigate the finite topological type of complete noncompact Riemannian manifolds with quadratic Ricci curvature decay, which generalizes a related result by Sha and Shen in 6 .
In 8 , we apply the volume comparison with asymptotically nonnegative Ricci curvature to investigate the corresponding topological results for manifolds with asymptotically nonnegative Ricci curvature.

International Journal of Mathematics and Mathematical Sciences
In this paper, we will extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to general radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number and finite topological type of manifolds with non-asymptotically almost nonnegative Ricci curvature.See Definitions 1.1 and 1.2 below for the notions of radially symmetric Ricci curvature lower bound, asymptotically almost nonnegative Ricci curvature, and non-asymptotically almost nonnegative Ricci curvature, resp.
Note that quadratic Ricci curvature decay is non-asymptotically almost nonnegative Ricci curvature, so our result is a generalization of the corresponding result of Lott and Shen in 7 mentioned above.See Theorem 1.7.
Definition 1.1.Let M be a complete n-Riemannian manifold n ≥ 2 , p ∈ M, and l : sup{d p, x | x ∈ M}.M has a radially symmetric Ricci curvature lower bound,k at the point p if there exists a continuous function k : 0, l → R such that, for any tangent vector v ∈ T x M radial from the point p, 1.1 One can refer to 9 for generalized space forms with radially symmetric curvature and the notion of tangent vector radial from a point.
The following is a volume comparison estimate for manifolds with general radially symmetric Ricci curvature lower bound, which is a generalization of that for manifolds with asymptotically nonnegative Ricci curvature and quadratic Ricci curvature decay by Zhu in 10 and Lott and Shen in 7 , respectively.Theorem 1.3.Let M be a complete n-Riemannian manifold n ≥ 2 with a radially symmetric Ricci curvature lower bound k : 0, l → R at the point p ∈ M, and let r ≤ R, s ≤ S, r ≤ s, R ≤ S, Γ be a measurable subset of the unit sphere in the tangent space T p M: Remark 1.4.The condition y > 0 on 0, l in 1.5 constitutes an extra assumption imposed on the unique solution y of 1.4 .In Theorem 1.3, we do not require that the radially symmetric Ricci curvature lower bound k : 0, l → R corresponds to the generalized space forms with radially symmetric curvature lower bound.Our purpose is to establish a volume comparison estimate effectively.
Applying the generalized volume comparison estimate, we can now investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.Theorem 1.5.Let M be a complete n-Riemannian manifold n ≥ 2 with non-asymptotically almost nonnegative Ricci curvature k : 0, l → R at the point p ∈ M, and let M be noncollapsing, that

International Journal of Mathematics and Mathematical
1.9 Theorem 1.6.Let M be a complete n-Riemannian manifold n ≥ 2 with non-asymptotically almost nonnegative Ricci curvature k : 0, l → R at the point p ∈ M, and M has weakly bounded geometry, that is, sec M ≥ −1, and 1 If there exist constants C, L > 0 such that the unique solution of 1.4 satisfies y t ≤ C 2 t L , then one has a constant C n, k, v > 0 depending only on n, k, and v such that, for r > 0, 1.10 2 If there exist constants C 1 , C 2 , K, L > 0 such that the unique solution of 1.5 satisfies Theorem 1.7.Let M be a complete n-Riemannian manifold n ≥ 2 with non-asymptotically almost nonnegative Ricci curvature k : 0, l → R at the point p ∈ M, sec x ≥ − C/d p, x α where C > 0, 0 ≤ α ≤ 2, and let M be non-collapsing, that is, , then M is of finite topological type with the additional assumption that for some constant C n, k, C, α, v > 0 depending only on n, k, C, α, and v.

A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound
Proof of Theorem 1.3.Choose polar coordinate r, θ at p. Define the function J r, θ by the formula dv M J n−1 dr dθ.

2.1
Then where cut θ is the distance from p to the cut point in direction θ.It is well known e.g., 11 that J satisfies the following:

2.3
Let y t be the unique solution of one of 1.4 and 1.5 Note that, by the uniqueness of the solution of ordinary differential equation, the solution of 1.4 always exists. .Then in the interval of y > 0, J y − y J ≤ 0, that is, J y − y J ≤ 0. By the initial condition of J and y, J y − y J ≤ 0. Thus, when y > 0, This shows that J/y is nonincreasing in the interval of y > 0.
Note that in the interval of J > 0 we must have y > 0. Thus it suffices to consider that y t is the unique solution of 1.4 .

2.5
Let t → t 0 , then J t 0 ≤ y t 0 0. This is a contradiction.
Thus consider the following lemma.
Lemma 2.1 see 12 .Let f, g be positive functions on 0, ∞ ; if f/g is nonincreasing, then for all R > r > 0, S > s > 0, s > r, S > R, one has where the last equality is due to

2.8
Then by integration on Γ, we have

Proof of Theorem 1.5
Proof of Theorem 1.5.Note that for r ≥ 3 there exists a point q ∈ S p r such that And since M does not collapse at infinity, that is, inf Thus, for r ≥ 3, by Theorem 1.3 2 , there is some constant C n, k, v such that vol B p r ≥ C n, k, v r 1− n−1 K−L .And note that for 1 < r ≤ 3, Theorem 1.5 is obtained.

Proof of Theorem 1.6
First let us recall Gromov's theorems 2 ; one can refer to 13 for the details.
Theorem 4.1 see 2 .Let M be an n-dimensional complete Riemannian manifold with sectional curvature K ≥ −1.Then there is a constant C n > 1 depending only on n such that, for any 0 < < 1 and any bounded subset where T X denotes the -neighborhood of X in M.
Theorem 4.2 see 2 .Let M be an n-dimensional complete Riemannian manifold and let p ∈ M.
For any fixed numbers r > 0 and r 0 ≤ 7 −n−1 , let B 0 j : B p j , r 0 , j 1, . . ., N, be a ball covering of B p, r with p j ∈ B p, r .Let B k j : 7 k B 0 j : B p j , 7 k r 0 , k 0, . . ., n 1.
where t is the smallest number such that, each ball B n j intersects at most t other balls B n j .
Proof of Theorem 1.6.By Theorem 4.1, there is a constant C n depending only on n such that for all balls B x, r with radius r ≤ 1 in M, Take r 0 7 −n−1 , and let B p j , 1/2 r 0 , j 1, . . ., N, be a maximal set of disjoint balls with p j ∈ B p, r , and let B k j , j 1, . . ., N, k 0, . . ., n 1, be the same as in Theorem 4.2.Then B 0 j , j 1, . . ., N, is a covering of B p, r .And let t, N be the same as in Theorem 4.2.

International Journal of Mathematics and Mathematical Sciences
If there exist constants C, L > 0 such that the unique solution of 1.4 satisfies y t ≤ C 2 t L , choosing Γ S n−1 p , s 0 in Theorem 1.3 1 , then, for S ≥ 0,

4.4
Then by the assumption that inf

4.5
Since each ball B k j has radius ≤ 1, it follows from 4.3 and Theorem 4.2 that If there exist constants C 1 , C 2 , K, L > 0 such that the unique solution of 1.5 satisfies

4.7
Similar to the above, there exists a constant C n, k, vol S p 1 , vol B p 1 > 0 such that

Proof of Theorem 1.7
We use critical point theory of the distance function to prove Theorem 1.7.
First of all, we recall some concepts cf., e.g., 3, 7, 14 .Notice that the distance function d p x : d p, x is not a smooth function on the cutlocus of p .Hence the critical points of d p are not defined in a usual sense.The notion of critical points of d p is introduced by Grove and Shiohama 15 .
A point x ∈ M is called a critical point of d p if for any unit vector v ∈ T x M there is a minimizing geodesic σ from x to p such that ∠ σ 0 , v ≤ π/2.
For every r, the open set M \ B p, r contains only finitely many unbounded components, U r .Each U r has finitely many boundary components, Σ r ⊂ ∂B p, r .In particular, Σ r is a closed subset.Let us say that a connected component Σ r of S p, r is good if it is part of the boundary of an unbounded component of M \ B p, r and there is a ray from p passing through Σ r .
Now we can introduce the following lemma.where then M is of finite topological type.As i in the proof of Theorem 1.1 in 8 , choose a good connected component Σ r of S p, r , for any x ∈ Σ r , and a ray γ from p passing through Σ r , choose q γ t such that t ≥ 2d p, x , and suppose that x is a critical point of d p , then e pq x ≥ δ 1 d p, x α/2 .

5.4
On the other hand, by the triangle inequality, For r large enough, by the assumption on the diameter growth, this is a contradiction.Thus, there does not exist a critical point of d p, • on any good connected component.By Lemma 5.1, M is of finite topological type.
ii Given that r > 0, choose a good connected component Σ r , of the boundary of an unbounded component of M \ B p, r .For any x, y ∈ Σ r , there is a continuous curve c : 0, s → Σ r from x to y. Suppose that d x, y > 2. Then there is a partition 0 t 0 < t 1 < • • • < t k r such that {B c t i , 1 } k i 0 are disjoint and B c t i , 2 B c t i 1 , 2 / ∅.Note that B c t i , 1 ⊂ B p, r 1 − B p, r − 1 .Thus  Then by i , Theorem 1.7 is obtained.

2 . 14 International 1 vol B p r − 1 − B p 1 ≤ r 1 r
Journal of Mathematics and Mathematical Sciences 2 Choose Γ S n−1 p ; for r ≥ 3, an easy computation shows that vol B p r 1 − B p r −

e
pq x ≤ 2D p, r , B c t i , 1 ≤ vol B p, r 1 − B p, r − 1 , i , c t i 1 ≤ C n, k, v vol B p r 1 − B p r − 1 .
Theorem 1.3 2 , there is a constant C n, k, C, α, v such that if the volume growth satisfies lim sup r → ∞ vol B p r r 1 α/2 − n−1 L−K < C n, k, C, α, v 5

4 or 1.5 satisfies C 1 t K ≤ y t ≤ C 2 t L , then one has a constant C n, k > 0 depending only on n and k such that
for any minimal geodesic γ from p to x .
Lemma 5.1 see Lemma 3.2, 7 ; cf., also 14 .Suppose that there is a r 0 > 0 such that if r > r 0 then there is no critical point of d p on any good component Σ r of S p, r .Then M has finite topological type.
Another concept is the diameter growth function D p, r .