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

Authors: Zisheng Hu; Yadong Jin; Senlin Xu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

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.


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## Body

## 1. 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.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 Definitions1.1 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.
LetM 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∈TxM radial from the point p,(1.1)Ric(v)≥(n-1)k(d(p,x)).One can refer to [9] for generalized space forms with radially symmetric curvature and the notion of tangent vector radial from a point.Definition 1.2.
LetM a complete noncompact n-Riemannian manifold (n≥2),p∈M,k:[0,∞)→R be a continuous positive function, and limt→∞k(t)=0.(i)
M has almost nonnegative Ricci curvature if Ric(x)≥(n-1)k(d(p,x)).Furthermore, (ii)
M has asymptotically nonnegative Ricci curvature if Ric(x)≥(n-1)k(d(p,x)) and C(k):=∫0∞tk(t)dt<∞.(iii)
M has non-asymptotically almost nonnegative Ricci curvature if Ric(x)≥(n-1)k(d(p,x)) and C(k):=∫0∞tk(t)dt=∞.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.
LetM 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  TpM:
(1.2)Ar,RΓ(p):={x∈M∣r≤d(p,x)≤R,γ̇(0)∈Γforanyminimalgeodesicγfromptox}.
Then
(1.3)vol(As,SΓ(p))vol(Ar,RΓ(p))≤∫sSyn-1(t)dt∫rRyn-1(t)dt,
where y(t) is the unique solution of one of the following two equations:
(1.4)y′′-k(t)y=0,y(0)=0,y′(0)=1,(1.5)y′′-k(t)y=0,y(0)=0,y′(0)=1,y>0on(0,l).
In particular,(1) if y(t) is the unique solution of (1.4), then (1.6)vol(As,SΓ(p))≤vol(Γ)∫sSyn-1(t)dt.
if y(t) is the unique solution of (1.5), then
(1.7)vol(As,SΓ(p))≤vol(As,sΓ(p))yn-1(s)∫sSyn-1(t)dt.
(2) If there exist constants C1,C2,K,L>0(0≤L-K<1/(n-1)) such that the unique solution of (1.4) or (1.5) satisfies C1tK≤y(t)≤C2tL, then one has a constant C(n,k)>0 depending only on n and k such that (1.8)vol(Bp(r+1)-Bp(r-1))≤C(n,k)vol(Bp(r-1))(r-1)1-(n-1)(L-K).Remark 1.4.
The conditiony>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 non-asymptotically almost nonnegative Ricci curvature.Theorem 1.5.
LetM 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 is, infx∈Mvol(Bx(1))≥v>0. If there exist constants C1,C2,K,L>0(0≤L-K<1/(n-1)) such that, the unique solution of (1.4) or (1.5) satisfies C1tK≤y(t)≤C2tL, then one has a constant C(n,k)>0 depending only on n and k such that for r>1,
(1.9)vol(Bp(r))≥C(n,k,v)r1-(n-1)(L-K).Theorem 1.6.
LetM 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  infx∈Mvol(Bx(7-n-1/2))≥v>0.
(1) If there exist constants C,L>0 such that the unique solution of (1.4) satisfies y(t)≤C2tL, then one has a constant C(n,k,v)>0 depending only on n, k, and v such that, for r>0,(1.10)∑i=0nbi(p,r)≤C(n,k,v)(1+r)(n2-1)L+n+1.
(2) If there exist constants C1,C2,K,L>0 such that the unique solution of (1.5) satisfies C1tK≤y(t)≤C2tL, then one has a constant C(n,k,v,vol(Sp(1)), vol(Bp(1)))>0 depending only on n, k, v, vol(Sp(1)), and vol(Bp(1)) such that, for r>1,(1.11)∑i=0nbi(p,r)≤C(n,C,v,vol(Sp(1)),vol(Bp(1)))(1+r)(n2-1)L+n+1.Theorem 1.7.
LetM 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, infx∈Mvol(Bx(1))≥v>0. If there exist constants C1,C2,K,L>0(0≤L-K<(1/(n-1)) such that the unique solution of (1.4) or (1.5) satisfies C1tK≤y(t)≤C2tL, then M is of finite topological type with the additional assumption that
(1.12)lim supr→∞vol(Bp(r))r1+(α/2)-(n-1)(L-K)<C(n,k,C,α,v)
for some constant C̃(n,k,C,α,v)>0 depending only on n, k, C, α, and v.


## 2. A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound

Proof of Theorem1.3.
Choose polar coordinate(r,θ) at p. Define the function J(r,θ) by the formula
(2.1)dvM=Jn-1drdθ.
Then
(2.2)vol(As,SΓ(p))=∫Γdθ∫min{s,cut(θ)}min{S,cut(θ)}Jn-1dr,
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)J′′-kJ≤0,0≤t≤cut(θ),J(0)=0,J′(0)=1.
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 ofy>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, (2.4)(Jy)′=1y2(J′y-y′J)≤0.
This shows that J/y is nonincreasing in the interval of y>0.
Note that in the interval ofJ>0 we must have y>0. Thus it suffices to consider that y(t) is the unique solution of (1.4).
Otherwise, suppose thatt0>0 is the first point such that y>0 in (0,t0), y(t0)=0, and J>0 in (0,t0]. By J(0)=y(0)=0, J′(0)=y′(0)=1, J/y is non-increasing in (0,t0):(2.5)Jy(0):=limt→0J(t)y(t)=limt→0J′(t)y′(t)=1,Jy(t)≤Jy(0)=1,t∈(0,t0),J(t)≤y(t),t∈(0,t0).
Let t→t0, then J(t0)≤y(t0)=0. This is a contradiction.Thus consider the following lemma.Lemma 2.1 (see [12]).
Letf,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
(2.6)∫sSf(t)dt∫rRf(t)dt≤∫sSg(t)dt∫rRg(t)dt.We have(2.7)∫min{r,cut(θ)}min{S,cut(θ)}Jn-1(t,θ)dt∫min{r,cut(θ)}min{R,cut(θ)}Jn-1(t,θ)dt≤∫min{r,cut(θ)}min{S,cut(θ)}yn-1(t)dt∫min{r,cut(θ)}min{R,cut(θ)}yn-1(t)dt≤∫rmin{S,cut(θ)}yn-1(t)dt∫rmin{R,cut(θ)}yn-1(t)dt≤∫rSyn-1(t)dt∫rRyn-1(t)dt,(becauseR<S),
where the last equality is due to (2.8)∫rmin{S,cut(θ)}yn-1(t)dt∫rmin{R,cut(θ)}yn-1(t)dt={∫rcut(θ)yn-1(t)dt∫rcut(θ)yn-1(t)dt=1,whencut(θ)≤R≤S,∫rcut(θ)yn-1(t)dt∫rRyn-1(t)dt,whenR≤cut(θ)≤S,∫rSyn-1(t)dt∫rRyn-1(t)dt,whenR≤S≤cut(θ).
Then by integration on Γ, we have (2.9)vol(Ar,SΓ(p))vol(Ar,RΓ(p))≤∫rSyn-1(t)dt∫rRyn-1(t)dt.
Similarly, (2.10)vol(As,SΓ(p))vol(Ar,RΓ(p))≤∫sSyn-1(t)dt∫rRyn-1(t)dt.In particular, (1)(2.11)vol(As,SΓ(p))≤vol(Ar,RΓ(p))∫rRyn-1(t)dt∫sSyn-1(t)dt=∫Γ∫min{r,cut(θ)}min{R,cut(θ)}Jn-1(t,θ)dtdθ∫rRyn-1(t)dt∫sSyn-1(t)dt=∫Γ∫min{r,cut(θ)}min{R,cut(θ)}Jn-1(t,θ)dt∫rRyn-1(t)dtdθ∫sSyn-1(t)dt.LetR→r, then (2.12)vol(As,SΓ(p))≤∫ΓJn-1(r,θ)yn-1(r)dθ∫sSyn-1(t)dt.Wheny(t) is the unique solution of (1.4), let r→0; by J(0)=y(0)=0, J′(0)=y′(0)=1; then we have (2.13)vol(As,SΓ(p))≤vol(Γ)∫sSyn-1(t)dt.Wheny(t) is the unique solution of (1.5), let r=s, we have (2.14)vol(As,SΓ(p))≤∫ΓJn-1(s,θ)yn-1(s)dθ∫sSyn-1(t)dt=vol(As,sΓ(p))yn-1(s)∫sSyn-1(t)dt.(2) Choose Γ=Spn-1; for r≥3, an easy computation shows that (2.15)vol(Bp(r+1)-Bp(r-1))vol(Bp(r-1)-Bp(1))≤∫r-1r+1yn-1(t)dt∫1r-1yn-1(t)dt≤∫r-1r+1(C2tL)n-1dt∫1r-1(C1tK)n-1dt=C2n-1((n-1)K+1)C1n-1((n-1)L+1)·(r+1)(n-1)L+1-(r-1)(n-1)L+1(r-1)(n-1)K+1-1≤2C2n-1((n-1)K+1)C1n-1((n-1)L+1)·(r+1)(n-1)L+1-(r-1)(n-1)L+1(r-1)(n-1)K+1=C(n,C1,C2,K,L)((1+2r-1)(n-1)K+1·(r+1)(n-1)(L-K)-(r-1)(n-1)(L-K))≤C(n,C1,C2,K,L)((1+C(n,K)r-1)·(r+1)(n-1)(L-K)-(r-1)(n-1)(L-K))=C(n,C1,C2,K,L)·(r-1)(n-1)(L-K)·((1+C(n,K)r-1)·(r+1r-1)(n-1)(L-K)-1)≤C(n,C1,C2,K,L)·(r-1)(n-1)(L-K)·((1+C(n,K)r-1)·(1+C(n,K,L)r-1)-1)=C(n,C1,C2,K,L)·(r-1)(n-1)(L-K)·(C(n,K)+C(n,K,L)r-1+C(n,K)·C(n,K,L)(r-1)2)≤C(n,C1,C2,K,L)1(r-1)1-(n-1)(L-K).


## 3. Proof of Theorem1.5

Proof of Theorem1.5.
Note that forr≥3 there exists a point q∈Sp(r) such that (Bp(r+1)-Bp(r-1))⊇Bq(1); thus
(3.1)vol(Bp(r+1)-Bp(r-1))≥volBq(1).
And since M does not collapse at infinity, that is, infx∈Mvol(Bx(1))≥v>0, for r≥3, we have
(3.2)vol(Bp(r+1)-Bp(r-1))≥v.
Thus, for r≥3, by Theorem 1.3(2), there is some constant C(n,k,v) such that vol(Bp(r))≥C(n,k,v)r1-(n-1)(K-L). And note that for 1<r≤3,
(3.3)Bp(r)⊇Bp(1).
Theorem 1.5 is obtained.


## 4. Proof of Theorem1.6

First let us recall Gromov's theorems [2]; one can refer to [13] for the details.Theorem 4.1 (see [2]).
LetM 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<ϵ<1 and any bounded subset  X⊂M,
(4.1)∑k=0nbk(X,TϵX)≤(1+diam(X)ϵ-1)nC(n)1+dia(X),
where TϵX denotes the ϵ-neighborhood of X in M.Theorem 4.2 (see [2]).
LetM be an  n-dimensional complete Riemannian manifold and let  p∈M. For any fixed numbers  r>0 and  r0≤7-n-1, let  Bj0:=B(pj,r0),  j=1,…,N, be a ball covering of  B(p,r) with  pj∈B(p,r). Let  Bjk:=7kBj0:=B(pj,7kr0),  k=0,…,n+1. Then
(4.2)∑i=0nbi(B(p,r),B(p,r+1))≤(e-1)Ntnsup{∑i=0nbi(Bjk,5Bjk):0≤k≤n,1≤j≤N},
where t is the smallest number such that, each ball Bjn intersects at most t other balls Bj′n.Proof of Theorem1.6.
By Theorem4.1, there is a constant C(n) depending only on n such that for all balls B(x,r) with radius r≤1 in M,
(4.3)∑i=0nbi(B(x,r),B(x,5r))≤C1(n).
Taker0=7-n-1, and let B(pj,(1/2)r0), j=1,…,N, be a maximal set of disjoint balls with pj∈B(p,r), and let Bjk, j=1,…,N, k=0,…,n+1, be the same as in Theorem 4.2. Then Bj0, j=1,…,N, is a covering of B(p,r). And let t, N be the same as in Theorem 4.2.
If there exist constantsC,L>0 such that the unique solution of (1.4) satisfies y(t)≤C2tL, choosing Γ=Spn-1, s=0 in Theorem 1.3(1), then, for S≥0,(4.4)vol(Bp(S))≤vol(S1n-1)∫0S(C2tL)n-1dt=vol(S1n-1)C2(n-1)L+1S(n-1)L+1=:C(n,k)S(n-1)L+1.
Then by the assumption that infx∈Mvol(Bx(7-n-1/2))≥v>0,
(4.5)N≤vol(Bx(r+(r0/2)))minjvol(Bpj(r0/2))≤C(n,k)(r+(r0/2))(n-1)L+1v≤C(n,k)(r+1)(n-1)L+1vt≤vol(Bpj((2/7)+(r0/2)))minj′vol(Bpj′(r0/2))≤vol(Bp(r+(2/7)+(r0/2)))v≤C(n,k)(r+(2/7)+(r0/2))(n-1)L+1v≤C(n,k)(r+1)(n-1)L+1v.
Since each ball Bjk has radius ≤1, it follows from (4.3) and Theorem 4.2 that
(4.6)∑i=0nbi(Bp(r),M)≤∑i=0nbi(Bp(r),Bp(r+1))≤(e-1)(C(n,k)(r+1)(n-1)L+1v)n+1C(n)=:C(n,k,v)(1+r)((n-1)L+1)(n+1).
If there exist constantsC1,C2,K,L>0 such that the unique solution of (1.5) satisfies C1tK≤y(t)≤C2tL, choosing Γ=Spn-1, s=1 in Theorem 1.3(1), then, for S≥1, (4.7)vol(Bp(S)-Bp(1))≤vol(Sp(1))yn-1(1)∫1Syn-1(t)dt≤vol(Sp(1))C1∫1S(C2tL)n-1dt≤vol(Sp(1))C1C2(n-1)L+1S(n-1)L+1=:C(n,k,vol(Sp(1)))S(n-1)L+1,vol(Bp(S))≤C(n,k,vol(Sp(1)))S(n-1)L+1+vol(Bp(1))≤C(n,k,vol(Sp(1)),vol(Bp(1)))S(n-1)L+1.
Similar to the above, there exists a constantC(n,k,vol(Sp(1)),vol(Bp(1)))>0 such that(4.8)∑i=0nbi(Bp(r),M)≤C(n,k,vol(Sp(1)),vol(Bp(1)))(1+r)((n-1)L+1)(n+1).


## 5. Proof of Theorem1.7

We use critical point theory of the distance function to prove Theorem1.7.First of all, we recall some concepts (cf., e.g., [3, 7, 14]). Notice that the distance function dp(x):=d(p,x) is not a smooth function (on the cutlocus of p). Hence the critical points of dp are not defined in a usual sense. The notion of critical points of dp is introduced by Grove and Shiohama [15].A pointx∈M is called a critical point of dp if for any unit vector v∈TxM there is a minimizing geodesic σ from x to p such that ∠(σ′(0),v)≤π/2.For everyr, the open set M∖B(p,r)¯ contains only finitely many unbounded components, Ur. Each Ur has finitely many boundary components, Σr⊂∂B(p,r). In particular, Σr is a closed subset. Let us say that a connected component Σr 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 Σr.Now we can introduce the following lemma.Lemma 5.1 (see (Lemma3.2, [7]); cf., also [14]).
Suppose that there is ar0>0 such that if  r>r0 then there is no critical point of  dp on any good component  Σr of  S(p,r). Then  M has finite topological type.Another concept is thediameter growth function𝒟(p,r).Definition 5.2.
The diameter growth function𝒟(p,r) is defined by
(5.1)𝒟(p,r)=supΣrdiam(∑r),
where the supremum is taken over all good components Σt of St and the diameter is measured using the metric on M.Proof of Theorem1.7.
(i) We first show that if a complete noncompact Riemannian manifold satisfiesK(x)≥-C/d(p,x)α, where C>0, 0≤α≤2, and the following diameter growth condition (5.2)lim supr→∞𝒟(p,r)rα/2<δ12,
where
(5.3)δ1=max0<ϵ≤1/20{2ϵ-arccosh(cosh22αC1/2ϵ)2αC1/2>0}
then M is of finite topological type.
As (i) in the proof of Theorem1.1 in [8], choose a good connected component Σr of S(p,r), for any x∈Σr, and a ray γ from p passing through Σr, choose q=γ(t) such that t≥2d(p,x), and suppose that x is a critical point of dp, then(5.4)epq(x)≥δ1d(p,x)α/2.
On the other hand, by the triangle inequality,
(5.5)epq(x)≤2𝒟(p,r),
thus,
(5.6)𝒟(p,r)≥δ12d(p,x)α/2,
For r large enough, by the assumption on the diameter growth, this is a contradiction.
Thus, there does not exist a critical point ofd(p,·) on any good connected component. By Lemma 5.1, M is of finite topological type.
(ii) Given thatr>0, choose a good connected component Σr, of the boundary of an unbounded component of M∖B(p,r)¯. For any x,y∈Σr, there is a continuous curve c:[0,s]→Σr from x to y. Suppose that d(x,y)>2. Then there is a partition 0=t0<t1<⋯<tk=r such that {B(c(ti),1)}i=0k are disjoint and B(c(ti),2)⋂B(c(ti+1),2)≠∅. Note that B(c(ti),1)⊂B(p,r+1)-B(p,r-1)¯. Thus(5.7)(k+1)v≤∑i=0kvol(B(c(ti),1))≤vol(B(p,r+1)-B(p,r-1)¯),diam(∑r)≤∑i=0k-1d(c(ti),c(ti+1))≤C(n,k,v)vol(Bp(r+1)-Bp(r-1)¯).
Then, by Theorem 1.3(2), there is a constant C̃(n,k,C,α,v) such that if the volume growth satisfies
(5.8)lim supr→∞vol(Bp(r))r1+(α/2)-(n-1)(L-K)<C̃(n,k,C,α,v)
the diameter growth satisfies
(5.9)lim supr→∞𝒟(p,r)rα/2<δ12.
Then by (i), Theorem 1.7 is obtained.



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*Source: 758531-2010-04-27.xml*