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

Volume 2010, Article ID 758531,14pages doi:10.1155/2010/758531

Research Article

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

Zisheng Hu,

Yadong Jin,

and Senlin Xu

1School of Mathematics and Computational Science, Shenzhen University, Shenzhen, Guangdong 518060, China

2School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou, Jiangsu 213001, China

3Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Correspondence should be addressed to Zisheng Hu,[email protected] Received 3 July 2009; Revised 16 October 2009; Accepted 17 February 2010 Academic Editor: Mircea-Eugen Craioveanu

Copyrightq2010 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.

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 curvaturesee1, the upper bound of total Betti numbergrowthof Riemannian manifoldssee2–4, and the finite topological type of complete noncompact Riemannian manifolds with nonnegative Ricci curvature or quadratic Ricci curvature decaysee3,5,6.

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 in6.

In8, we apply the volume comparison with asymptotically nonnegative Ricci cur- vature 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 Definitions1.1and1.2below 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 in7mentioned above.SeeTheorem 1.7.

Definition 1.1. Let M be a complete n-Riemannian manifold n ≥ 2, p ∈ M, and l : sup{dp, x | x ∈ M}. M has a radially symmetric Ricci curvature lower bound,k at the pointpif there exists a continuous functionk :0, l → Rsuch that, for any tangent vector v∈TxMradial from the pointp,

Ricv≥n−1k d

p, x

. 1.1

One can refer to9for generalized space forms with radially symmetric curvature and the notion of tangent vector radial from a point.

Definition 1.2. LetM a complete noncompactn-Riemannian manifoldn ≥ 2,p ∈ M,k : 0,∞ → Rbe a continuous positive function, and limt→ ∞kt 0.

iMhas almost nonnegative Ricci curvature if Ricx≥n−1kdp, x.

Furthermore,

iiMhas asymptotically nonnegative Ricci curvature if Ricx≥n−1kdp, xand Ck:_∞

0 tktdt <∞.

iiiM has non-asymptotically almost nonnegative Ricci curvature if Ricx ≥ n− 1kdp, xandCk:_∞

0 tktdt∞.

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 10and Lott and Shen in7, respectively.

Theorem 1.3. LetMbe a completen-Riemannian manifoldn≥2with a radially symmetric Ricci curvature lower boundk :0, l → Rat the pointp∈M, and letr≤R,s≤S,r ≤s,R≤S,Γbe a measurable subset of the unit sphere in the tangent spaceTpM:

A^Γ_r,R p

:

x∈M|r≤d p, x

≤R,γ0˙ ∈Γfor any minimal geodesicγ fromp tox .

1.2

Then

vol A^Γ_s,S

p vol

A^Γ_r,R p ≤

_S

s yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt, 1.3

whereytis the unique solution of one of the following two equations:

y−kty0,

y0 0, y0 1, 1.4

y−kty0, y0 0, y0 1,

y >0 on 0, l.

1.5

In particular,1ifytis the unique solution of 1.4, then

vol A^Γ_s,S

p

≤volΓ ^S

s

yⁿ⁻¹tdt. 1.6

ifytis the unique solution of 1.5, then

vol A^Γ_s,S

p

≤ vol A^Γ_s,s

p yⁿ⁻¹s

S s

yⁿ⁻¹tdt. 1.7

2If there exist constantsC1, C2, K, L > 00 ≤ L−K < 1/n−1such that the unique solution of1.4or1.5satisfiesC₁t^K≤yt≤C₂t^L, then one has a constantCn, k>0 depending only onnandksuch that

vol

Bpr1−Bpr−1

≤Cn, k vol

Bpr−1

r−1^{1−n−1L−K}. 1.8

Remark 1.4. The conditiony >0 on0, lin1.5constitutes an extra assumption imposed on the unique solutionyof1.4. InTheorem 1.3, we do not require that the radially symmetric Ricci curvature lower boundk :0, l → Rcorresponds 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. LetMbe a completen-Riemannian manifoldn≥2with non-asymptotically almost nonnegative Ricci curvaturek : 0, l → R at the pointp ∈ M, and letMbe noncollapsing, that

is, inf_x∈MvolBx1≥ v > 0. If there exist constantsC1, C2, K, L > 00 ≤ L−K < 1/n−1 such that, the unique solution of 1.4or1.5satisfiesC₁t^K ≤yt≤C₂t^L, then one has a constant Cn, k>0 depending only onnandksuch that forr >1,

vol Bpr

≥Cn, k, vr^{1−n−1L−K}. 1.9

Theorem 1.6. LetMbe a completen-Riemannian manifoldn≥2with non-asymptotically almost nonnegative Ricci curvaturek:0, l → Rat the pointp∈M, andMhas weakly bounded geometry, that is, secM≥ −1, and inf_x∈MvolBx7⁻ⁿ⁻¹/2≥v >0.

1If there exist constantsC, L >0 such that the unique solution of 1.4satisfiesyt≤C₂t^L, then one has a constantCn, k, v>0 depending only onn,k,andvsuch that, forr >0,

n i0

b_i p, r

≤Cn, k, v1r^n2−1Ln1. 1.10

2If there exist constantsC₁, C₂, K, L > 0 such that the unique solution of 1.5satisfies C1t^K≤yt≤C2t^L, then one has a constantCn, k, v,volSp1, volBp1>0 depending only onn,k,v, volSp1,and volBp1such that, forr >1,

n i0

bi

p, r

≤C

n, C, v,vol Sp1

,vol

Bp1

1r^n2−1Ln1. 1.11

Theorem 1.7. LetMbe a completen-Riemannian manifoldn≥2with non-asymptotically almost nonnegative Ricci curvaturek : 0, l → R at the pointp ∈ M, secx ≥ −C/dp, x^αwhere C > 0, 0 ≤ α ≤ 2, and letMbe non-collapsing, that is, inf_x∈MvolBx1 ≥ v > 0. If there exist constantsC₁, C₂, K, L >0 0≤L−K <1/n−1such that the unique solution of 1.4or1.5 satisfiesC1t^K≤yt≤C2t^L, thenMis of finite topological type with the additional assumption that

lim sup

r→ ∞

vol B_pr

r1α/2−n−1L−K < Cn, k, C, α, v 1.12

for some constantCn, k, C, α, v >0 depending only onn,k,C,α,andv.

2. A Volume Comparison Estimate with

Radially Symmetric Ricci Curvature Lower Bound

Proof ofTheorem 1.3. Choose polar coordinate r, θ at p. Define the function Jr, θby the formula

dv_MJⁿ⁻¹dr dθ. 2.1

Then

vol A^Γ_s,S

p

Γdθ

min{S,cutθ}

min{s,cutθ} Jⁿ⁻¹dr, 2.2

where cutθis the distance frompto the cut point in directionθ. It is well knowne.g.,11 thatJsatisfies the following:

J−kJ≤0, 0≤t≤cutθ,

J0 0, J0 1. 2.3

Letytbe the unique solution of one of1.4and1.5 Note that, by the uniqueness of the solution of ordinary differential equation, the solution of1.4always exists..

Then in the interval ofy > 0,Jy−yJ ≤ 0, that is, Jy−yJ ≤ 0. By the initial condition ofJandy,Jy−yJ ≤0. Thus, wheny >0,

J y

1

y²

Jy−yJ

≤0. 2.4

This shows thatJ/yis nonincreasing in the interval ofy >0.

Note that in the interval ofJ >0 we must havey >0. Thus it suffices to consider that ytis the unique solution of1.4.

Otherwise, suppose thatt0 >0 is the first point such thaty >0 in0, t0,yt0 0, and J >0 in0, t0. ByJ0 y0 0,J0 y0 1,J/yis non-increasing in0, t0:

J

y0:lim

t→0

Jt yt lim

t→0

Jt yt 1, J

yt≤ J

y0 1, t∈0, t0, Jt≤yt, t∈0, t0.

2.5

Lett → t₀, thenJt0≤yt0 0. This is a contradiction.

Thus consider the following lemma.

Lemma 2.1see12. Letf, gbe positive functions on0,∞; iff/gis nonincreasing, then for allR > r >0,S > s >0,s > r,S > R, one has

_S

s ftdt _R

r ftdt ≤ _S

s gtdt _R

r gtdt. 2.6

We have

min{S,cutθ}

min{r,cutθ} Jⁿ⁻¹t, θdt min{R,cutθ}

min{r,cutθ} Jⁿ⁻¹t, θdt ≤

min{S,cutθ}

min{r,cutθ} yⁿ⁻¹tdt min{R,cutθ}

min{r,cutθ} yⁿ⁻¹tdt

≤

min{S,cutθ}

r yⁿ⁻¹tdt

min{R,cutθ}

r yⁿ⁻¹tdt

≤ _S

r yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt, becauseR < S,

2.7

where the last equality is due to

min{S,cutθ}

r yⁿ⁻¹tdt

min{R,cutθ}

r yⁿ⁻¹tdt

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ _cutθ

r yⁿ⁻¹tdt _cutθ

r yⁿ⁻¹tdt 1, when cutθ≤R≤S, _cutθ

r yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt , whenR≤cutθ≤S, _S

r yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt, whenR≤S≤cutθ.

2.8

Then by integration onΓ, we have

vol A^Γ_r,S

p vol

A^Γ_r,R p ≤

_S

r yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt. 2.9

Similarly,

vol A^Γ_s,S

p vol

A^Γ_r,R p ≤

_S

s yⁿ⁻¹tdt _R

r yⁿ⁻¹tdt. 2.10

In particular,1

vol A^Γ_s,S

p

≤ vol A^Γ_r,R

p _R

r yⁿ⁻¹tdt

S s

yⁿ⁻¹tdt

Γ

min{R,cutθ}

min{r,cutθ} Jⁿ⁻¹t, θdt dθ _R

r yⁿ⁻¹tdt

S s

yⁿ⁻¹tdt

Γ

min{R,cutθ}

min{r,cutθ} Jⁿ⁻¹t, θdt _R

r yⁿ⁻¹tdt dθ

S s

yⁿ⁻¹tdt.

2.11

LetR → r, then

vol A^Γ_s,S

p

≤

Γ

Jⁿ⁻¹r, θ yⁿ⁻¹r dθ

S s

yⁿ⁻¹tdt. 2.12

Whenytis the unique solution of1.4, letr → 0; byJ0 y0 0,J0 y0 1; then we have

vol A^Γ_s,S

p

≤volΓ ^S

s

yⁿ⁻¹tdt. 2.13

Whenytis the unique solution of1.5, letrs, we have

vol A^Γ_s,S

p

≤

Γ

Jⁿ⁻¹s, θ yⁿ⁻¹s dθ_S

s yⁿ⁻¹tdt

vol A^Γ_s,s

p yⁿ⁻¹s

_S

s yⁿ⁻¹tdt.

2.14

2ChooseΓ Sⁿ⁻¹_p ; forr≥3, an easy computation shows that vol

Bpr1−Bpr−1 vol

B_pr−1−B_p1

≤ _r1

r−1yⁿ⁻¹tdt _r−1

1 yⁿ⁻¹tdt

≤ _r1

r−1

C₂t^L_{n−1 r−1} dt

1

C₁t^K_n−1 dt Cⁿ⁻¹₂ n−1K1

Cⁿ⁻¹₁ n−1L1 ·r1^n−1L1−r−1^n−1L1 r−1^n−1K1−1

≤ 2C₂ⁿ⁻¹n−1K1

C₁ⁿ⁻¹n−1L1 ·r1^n−1L1−r−1^n−1L1 r−1^n−1K1 Cn, C1, C₂, K, L

1 2 r−1

_n−1K1

·r1^n−1L−K−r−1^n−1L−K

≤Cn, C1, C2, K, L

1Cn, K r−1

·r1^n−1L−K−r−1^n−1L−K

Cn, C1, C2, K, L·r−1^n−1L−K·

1 Cn, K r−1

· r1

r−1

_n−1L−K

−1

≤Cn, C1, C2, K, L·r−1^n−1L−K·

1Cn, K r−1

·

1Cn, K, L r−1

−1

Cn, C1, C2, K, L·r−1^n−1L−K·

Cn, K Cn, K, L

r−1 Cn, K·Cn, K, L r−1²

≤Cn, C1, C₂, K, L 1

r−1^{1−n−1L−K}.

2.15

3. Proof of Theorem 1.5

Proof ofTheorem 1.5. Note that forr ≥ 3 there exists a pointq ∈Sprsuch thatBpr1− B_pr−1⊇B_q1; thus

vol

Bpr1−Bpr−1

≥volBq1. 3.1

And sinceMdoes not collapse at infinity, that is, inf_x∈MvolBx1≥v >0, forr≥3, we have vol

Bpr1−Bpr−1

≥v. 3.2

Thus, forr ≥ 3, byTheorem 1.32, there is some constantCn, k, vsuch that volBpr ≥ Cn, k, vr^{1−n−1K−L}. And note that for 1< r≤3,

Bpr⊇Bp1. 3.3

Theorem 1.5is obtained.

4. Proof of Theorem 1.6

First let us recall Gromov’s theorems2; one can refer to13for the details.

Theorem 4.1see2. LetMbe an n-dimensional complete Riemannian manifold with sectional curvatureK≥ −1. Then there is a constantCn>1 depending only onnsuch that, for any 0< <1 and any bounded subsetX ⊂M,

n k0

bkX, TX≤

1diamX⁻¹_n

Cn^1diaX, 4.1

whereTXdenotes the-neighborhood ofXinM.

Theorem 4.2see2. LetMbe ann-dimensional complete Riemannian manifold and letp∈M.

For any fixed numbersr >0 andr0 ≤ 7⁻ⁿ⁻¹, letB_j⁰ :Bpj, r0,j 1, . . . , N, be a ball covering of Bp, rwithp_j∈Bp, r. LetB_j^k:7^kB⁰_j :Bpj,7^kr₀,k0, . . . , n1. Then

n i0

b_i B

p, r , B

p, r1

≤e−1Ntⁿsup _n

i0

b_i

B^k_j,5B_j^k

: 0≤k≤n, 1≤j≤N

, 4.2

wheretis the smallest number such that, each ballBⁿ_j intersects at mosttother ballsBⁿ_j.

Proof ofTheorem 1.6. ByTheorem 4.1, there is a constantCndepending only onnsuch that for all ballsBx, rwith radiusr ≤1 inM,

n i0

biBx, r, Bx,5r≤C1n. 4.3

Taker0 7⁻ⁿ⁻¹, and letBpj,1/2r0,j 1, . . . , N, be a maximal set of disjoint balls withp_j∈Bp, r, and letB^k_j,j 1, . . . , N,k0, . . . , n1, be the same as inTheorem 4.2. Then B⁰_j,j1, . . . , N, is a covering ofBp, r. And lett,Nbe the same as inTheorem 4.2.

If there exist constantsC, L > 0 such that the unique solution of1.4satisfiesyt ≤ C₂t^L, choosingΓ Sⁿ⁻¹_p ,s0 inTheorem 1.31, then, forS≥0,

vol BpS

≤vol

Sⁿ⁻¹₁ ^S

0

C2t^L_n−1 dt vol

Sⁿ⁻¹₁ C2

n−1L1S^n−1L1 :Cn, kS^n−1L1.

4.4

Then by the assumption that inf_x∈MvolBx7⁻ⁿ⁻¹/2≥v >0,

N≤ volBxr r₀/2 min_jvol

B_pjr0/2

≤ Cn, kr r0/2^n−1L1 v

≤ Cn, kr1^n−1L1 v

t≤ vol

B_pj2/7 r0/2 minjvol

Bp_jr0/2

≤ vol

Bpr 2/7 r0/2 v

≤ Cn, kr 2/7 r0/2^n−1L1 v

≤ Cn, kr1^n−1L1

v .

4.5

Since each ballB^k_j has radius≤1, it follows from4.3andTheorem 4.2that

n i0

b_i

B_pr, M

≤ⁿ

i0

b_i

B_pr, Bpr1

≤e−1

Cn, kr1^n−1L1 v

_n1 Cn :Cn, k, v1r^n−1L1n1.

4.6

If there exist constantsC1, C2, K, L > 0 such that the unique solution of1.5satisfies C₁t^K≤yt≤C₂t^L, choosingΓ Sⁿ⁻¹_p ,s1 inTheorem 1.31, then, forS≥1,

vol

B_pS−B_p1

≤ vol Sp1 yⁿ⁻¹1

S 1

yⁿ⁻¹tdt

≤ vol S_p1 C₁

S 1

C₂t^L_n−1 dt

≤ vol Sp1 C1

C₂

n−1L1S^n−1L1 :C

n, k,vol S_p1

S^n−1L1, vol

B_pS

≤C

n, k,vol

S_p1

S^n−1L1vol B_p1

≤C

n, k,vol S_p1

,vol

B_p1

S^n−1L1.

4.7

Similar to the above, there exists a constant Cn, k,volSp1,volBp1 > 0 such that

n i0

b_i

B_pr, M

≤C

n, k,vol S_p1

,vol

B_p1

1r^n−1L1n1. 4.8

5. Proof of Theorem 1.7

We use critical point theory of the distance function to proveTheorem 1.7.

First of all, we recall some concepts cf., e.g., 3, 7, 14. Notice that the distance functiondpx : dp, xis not a smooth functionon the cutlocus ofp. Hence the critical points ofd_pare not defined in a usual sense. The notion of critical points ofd_pis introduced by Grove and Shiohama15.

A pointx ∈Mis called a critical point ofdpif for any unit vectorv∈ TxMthere is a minimizing geodesicσfromxtopsuch that∠σ0, v≤π/2.

For every r, the open set M \ Bp, r contains only finitely many unbounded components, U_r. Each U_r has finitely many boundary components, Σr ⊂ ∂Bp, r. In particular,Σr is a closed subset. Let us say that a connected componentΣrofSp, ris good if it is part of the boundary of an unbounded component ofM\Bp, rand there is a ray from ppassing throughΣr.

Now we can introduce the following lemma.

Lemma 5.1seeLemma 3.2,7; cf., also14. Suppose that there is ar0>0 such that ifr > r0

then there is no critical point ofd_pon any good componentΣrofSp, r. ThenMhas finite topological type.

Another concept is the diameter growth functionDp, r.

Definition 5.2. The diameter growth functionDp, ris defined by

D p, r

sup

Σr

diam

r

, 5.1

where the supremum is taken over all good componentsΣtofS_tand the diameter is measured using the metric onM.

Proof ofTheorem 1.7. iWe first show that if a complete noncompact Riemannian manifold satisfiesKx ≥ −C/dp, x^α, whereC > 0, 0 ≤ α ≤ 2, and the following diameter growth condition

lim sup

r→ ∞

D p, r r^α/2 < δ1

2 , 5.2

where

δ1 max

0<≤1/20

⎧⎨

⎩2−arc cosh

cosh²2^αC^1/2 2^αC^1/2 >0

⎫⎬

⎭ 5.3

thenMis of finite topological type.

Asiin the proof of Theorem 1.1 in8, choose a good connected componentΣr of Sp, r, for anyx ∈ Σr, and a rayγ fromp passing throughΣr, choose q γtsuch that t≥2dp, x, and suppose thatxis a critical point ofd_p, then

epqx≥δ1d p, x_α/2

. 5.4

On the other hand, by the triangle inequality, e_pqx≤2D

p, r

, 5.5

thus,

D p, r

≥ δ₁ 2 d

p, xα/2

, 5.6

Forrlarge enough, by the assumption on the diameter growth, this is a contradiction.

Thus, there does not exist a critical point ofdp,·on any good connected component.

ByLemma 5.1,Mis of finite topological type.

ii Given that r > 0, choose a good connected component Σr, of the boundary of an unbounded component of M \ Bp, r. For any x, y ∈ Σr, there is a continuous curve c : 0, s → Σr from x to y. Suppose that dx, y > 2. Then there is a partition

0t0< t1<· · ·< tkr such that{Bcti,1}^k_i0 are disjoint andBcti,2

Bcti1,2/∅.

Note thatBcti,1⊂Bp, r1−Bp, r−1. Thus k1v≤^k

i0

volBcti,1≤vol B

p, r1

−B

p, r−1 ,

diam

r

≤^k−1

i0

dcti, cti1≤Cn, k, vvol

Bpr1−Bpr−1 .

5.7

Then, byTheorem 1.32, there is a constantCn, k, C, α, v such that if the volume growth satisfies

lim sup

r→ ∞

vol Bpr

r1α/2−n−1L−K <Cn, k, C, α, v 5.8

the diameter growth satisfies

lim sup

r→ ∞

D p, r r^α/2 < δ1

2 . 5.9

Then byi,Theorem 1.7is obtained.

Acknowledgment

The authors would like to thank the referee for the comments and suggestions.

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