Mathematics
September 2021 Vol. 64 No. 9: 1937–1950 https://doi.org/10.1007/s11425-021-1869-5
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ARTICLES.
Relative volume comparison of Ricci flow
Gang Tian
1& Zhenlei Zhang
2,∗1School of Mathematical Sciences and Beijing International Center for Mathematical Research, Peking University, Beijing100871, China;
2School of Mathematical Sciences, Capital Normal University, Beijing100048, China Email: gtian@math.pku.edu.cn, zhleigo@aliyun.com
Received January 3, 2021; accepted April 6, 2021; published online May 19, 2021
Abstract In this paper we derive a relative volume comparison of Ricci flow under a certain local curvature condition. It is a refinement of Perelman’s no local collapsing theorem in Perelman (2002).
Keywords Ricci flow, relative volume comparison, local entropy MSC(2020) 53E20, 53C21
Citation: Tian G, Zhang Z L. Relative volume comparison of Ricci flow. Sci China Math, 2021, 64: 1937–1950, https://doi.org/10.1007/s11425-021-1869-5
1 Introduction
In this paper we establish a relative volume comparison of Ricci flow (see Theorem 1.2 below). It is a refinement of Perelman’s no local collapsing estimate in [14]. The major advantage of our volume estimate is that, like the Bishop-Gromov relative volume comparison theorem on a static manifold, it can be applied to Ricci flow with a collapsing structure.
In a subsequent joint paper with Song [18], we will use it to study the Gromov-Hausdorff convergence of K¨ahler-Ricci flow on a K¨ahler manifold with a semi-ample canonical line bundle.
Ricci flow was introduced by Hamilton [8] in 1982. It is an evolution of Riemannian metricsg(t) on a manifold in the sense that
∂
∂tg(t) =−2 Ric(g(t)). (1.1)
The no local collapsing theorem of Perelman is a crucial step in his work of Ricci flow on three-manifold (see [14, 15]). It is applied repeatedly in the two papers whenever a Gromov-Hausdorff convergence is taken. Perelman introduced two approaches to the no local collapsing theorem: the W-entropy and reduced volume (see the corresponding Theorems 4.1 and 8.2 in [14], respectively). The latter theorem is conceptionally weaker; however, it is much more powerful in application since it holds in a uniform way.
The theorem can be stated as follows.
* Corresponding author
Theorem 1.1 (See [14]). For any A > 0 and dimension m there exists κ = κ(m, A) >0 with the following property. LetM be a closedm-manifold andx0 be a base point. Ifg(t),0tr20, is a smooth solution to the Ricci flow on M such that
|Rm|(x, t)r0−2 for allx∈Bg(0)(x0, r0), t∈[0, r02], (1.2) and the volume of the metric ball Bg(0)(x0, r0)at timet= 0 satisfies
volg(0)(Bg(0)(x0, r0))A−1rm0, (1.3) then for any other metric ball Bg(r2
0)(y, r)⊂Bg(r20)(x0, Ar0)withrr0 and
|Rm|(x, t)r−2 for all x∈Bg(r2
0)(y, r), t∈[r02−r2, r02], (1.4) one has
volg(r20)(Bg(r20)(x, r))κrm. (1.5) In order to prove the theorem, Perelman used the monotonicity of the reduced volume under the Ricci flow which was introduced by himself [14]. In [22], Zhang localized Perelman’s W-entropy and the differential Harnack inequality, and proved a uniform local Sobolev inequality for the metric ball Bg(r20)(x0, Ar0) (see [22, Theorem 6.3.2]). As a direct consequence, he showed that, in order to estimate the volume of a metric ball at timet=r20 (see (1.5)), instead of assuming the curvature condition (1.4) on a parabolic domain, one just needs the scalar curvature estimate at the time slicet=r02, namely,
R(x, r20)r0−2 for allx∈Bg(r20)(x, r). (1.6) Recently, Wang [19] gave an independent proof of this improved no local collapsing theorem and found its application to K¨ahler-Ricci flow on smooth minimal models of general type.
Our main result is the following relative version of Perelman’s Theorem 1.1. The key improvement is the removal of the non-collapsing condition (1.3) for the initial metric.
Theorem 1.2. For any A > 0 and dimension m there exists κ = κ(m, A) > 0 with the following property. Let M be a closed m-manifold andx0 be a base point. If g(t),0tr20, is a smooth solution to the Ricci flow onM such that
|Ric|r−02 onBg(r2
0)(x0, r0)×[0, r20], (1.7) then for any metric ball Bg(r20)(x, r)⊂Bg(r20)(x0, Ar0)satisfying
R(·, r02)r−2 inBg(r20)(x, r), (1.8) one has the relative volume comparison at timer20
volg(r20)(Bg(r20)(x, r))
rm κ·volg(r20)(Bg(r20)(x0, r0))
r0m . (1.9)
We shall also use the localizedW-entropy in the proof of the theorem. Two new ingredients are also necessary. The first one is the “equivalence” of the volume ratio of a metric ball and the local entropy of the ball. It gives the sharp geometric interpretation of the local entropy. In the previous works [14,19,22], the non-collapsing condition, such as (1.3), is essentially involved in order to estimate the lower bound of the entropy. This is the leading observation that motivates the theorem. Another ingredient is the local Li-Yau type lower bound of the conjugate heat kernel. As in the usual case on a static manifold, the Li- Yau heat kernel lower bound that involves the volume of the metric ball of the radius√
tis much sharper than the heat kernel lower bound of the form t−m/2e−dct2. Actually, the conjugate heat kernel lower bound used in [19,22] is basically given by the reduced volume as showed by Perelman [14, Corollary 9.5].
The lower bound in [14, 19, 22] has the form exactly as t−m/2e−d2ct which is too weak to be used in the collapsing case. The proof of the Li-Yau lower bound of the conjugate heat kernel of Ricci flow is the main technical part of the paper.
Remark 1.3. Because the Ricci curvature is uniformly bounded on the space-time domain Bg(r20)(x0, r0)×[0, r02], the volumes of metric balls Bg(0)(x0,e−1r0) and Bg(r20)(x0, r0) are comparable under Ricci flow. So the relative volume comparison (1.9) is equivalent to
volg(r2 0)(Bg(r2
0)(x, r))
rm κ(m, A)·volg(0)(Bg(0)(x0,e−1r0))
rm0 (1.10)
which gives the volume comparison at timest= 0 andt=r20.
We end the introduction with a brief discussion on the organization of the paper. In Sections 2–4 we prove the main Theorem 1.2. The key ingredient is the Li-Yau type lower bound of the conjugate heat kernel under Ricci flow. The proof uses the comparison principle of Cheeger and Yau [6] on the heat kernel and the Harnack inequalities under Ricci flow: Perelman’s Harnack inequality [14] for heat equations and Kuang-Zhang’s Harnack inequality [11] for conjugate heat equations.
2 Local entropy, volume ratio and Ricci flow
In [14], Perelman introduced the entropy functional on a Riemannian manifold and proved its monotonic- ity under Ricci flow, and then he applied it to prove the no local collapsing Theorem 4.1 in [14]. Our aim is to prove a relative version of Perelman’s no local collapsing theorem. In this section we first recall the basic notations of the local entropy and prove the crucial fact that, under a certain curvature condition, the local entropy is equivalent to the volume ratio. Then we prove the partial monotonicity of the volume ratio under the Ricci flow with an error term involving a solution to the conjugate heat equation.
Let (M, g) be a compact Riemannianm-manifold. Recall Perelman’sW-functional (see [14]) W(g, f, τ) =
ˆ
M
[τ(R+|∇f|2) +f −m](4πτ)−m/2e−fdv. (2.1) After putting u= (4πτ)−m/4e−f /2, we can rewrite it as
W(g, u, τ) =τ ˆ
M
(Ru2+ 4|∇u|2)dv− ˆ
M
u2logu2dv−m
2 log(4πτ)−m. (2.2) Let Ω be any bounded domain of M. We define the local entropy (see [19, 22])
μΩ(g, τ) = inf
W(g, u, τ)
u∈C0∞(Ω), ˆ
Ω
u2dv= 1
. (2.3)
When Ω = M it is exactly the entropy μ(g, τ) introduced in Perelman’s paper [14]. It can be easily checked that the entropy satisfies the scaling invariance
μΩ(cg, cτ) =μΩ(g, τ) (2.4)
for any positive constantc. Moreover, the entropy satisfies the monotonicity
μΩ(g, τ)μΩ(g, τ) (2.5)
for any subdomain Ω⊂Ω. When Ω has a smooth boundary, there always exists a minimizer ofμΩwhich is smooth in Ω and continuous up to the boundary (see [16]).
Following Perelman [14], we also define the local energy functional, for anya >0, λa,Ω(g) = inf
ˆ
Ω
(Ru2+a|∇u|2)dv
u∈C0∞(Ω), ˆ
Ω
u2dv= 1
. (2.6)
It is obvious that λa,Ω is the smallest eigenvalue of the operator R−aΔ with Dirichlet condition. It satisfies the scaling propertyλa,Ω(cg) =c−1λa,Ω(g).
2.1 Local entropy and volume ratio
The entropy μΩis roughly equivalent to the log-Sobolev inequality of the domain Ω. We will show that it is also crucially related to the volume ratio of the domain. For some other estimates ofμΩ, please refer to [19, 22].
Lemma 2.1. For any domainΩ with a boundary, metricg andτ >0 we have μΩ(g, τ)τ λ4,Ω+ log volg(Ω)−m
2 logτ+C(m) (2.7)
and
μΩ(g, τ)τ λ3,Ω+ log volg(Ω)−mlogCs(Ω)−C(m), (2.8) where Cs(Ω)is the Sobolev constant of Ωin the sense that
Ω
fm−22m dv m−22m
Cs
Ω
|∇f|2dv 1/2
, ∀f ∈C0∞(Ω). (2.9)
Proof. We adopt the arguments from [24]. We first prove the upper bound. Letλa =λa,Ωfor simplicity.
Letube the eigenfunction ofλ4. Then by the definition ofμΩ and the trivial fact−xlogx1 for any x >0, we have
μΩ(g, τ)τ λ4(g) + volg(Ω)−m
2 logτ+C(m), ∀τ >0.
Then we apply the scaling invariance ofλandμΩto get μΩ(cg, cτ)cτ λ4(cg) + volcg(Ω)−m
2 log(cτ) +C(m)
=τ λ4(g) +cm/2·volg(Ω)−m
2 logτ−m
2 logc+C(m).
In particular if we choosec= volg(Ω)−2/m, then we get
μ(g, τ) =μ(cg, cτ)τ λ4(g) + log volg(Ω)−m
2 logτ+C(m).
Then we prove the lower bound. We shall apply the Sobolev inequality and the Jensen inequality: for any u∈C0∞(Ω) with´
Ωu2dv= 1, we have, with respect to the measuredμ=u2dv,
− ˆ
Ω
u2logu2dv=− ˆ
Ω
logu2dμ=−m−2 2
ˆ
Ω
logum−22m −2dμ
−m−2
2 log ˆ
Ω
um−22m −2dμ
=−mlog
ˆ
Ω
um2m−2dv m−22m
=−mlog
Ω
um−22m dv m2m−2
−m−2
2 log vol(Ω) −mlogCs−m
2 log
Ω
|∇u|2dv−m−2
2 log vol(Ω)
=−mlogCs−m 2 log
ˆ
Ω
|∇u|2dv+ log vol(Ω).
Thus,
W(g, u, τ)τ ˆ
Ω
(Ru2+ 4|∇u|2)dv−m 2 log
ˆ
Ω
|∇u|2dv−mlogCs+ log vol(Ω)−m
2 logτ−C(m) τ λ3+τ
ˆ
Ω
|∇u|2dv−m 2 log
τ
ˆ
Ω
|∇u|2dv
+ log vol(Ω)−mlogCs−C(m).
Finally we use the easy fact
x−m
2 logx1 2 +m
2 log 2, ∀x >0 to conclude that
W(g, u, τ)τ λ3+ log vol(Ω)−mlogCs−C(m).
It gives the lower bound ofμΩ(g, τ).
Corollary 2.2. Let B(x,2r)⊂M be a metric ball with∂B(x,2r)=∅. If
Cs(B(x, r))C(m)·r (2.10)
and
R−mr−2 inB(x, r), (2.11)
then
logvolg(B(x, r))
rm inf
0<τ2r2μB(x,r)(g, τ) +C(m). (2.12) Proof. The eigenvalueλ3,B(x,r)admits the trivial lower bound
λ3,B(x,r) inf
B(x,r)R−mr−2. (2.13)
Substituting the estimate into the formula (2.8) gives the desired result.
Remark 2.3. The Sobolev constant estimate (2.10) can be derived from some curvature condition, such as Ric−(n−1)r−2in B(x,2r) or some integral Ricci curvature condition onB(x,2r).
Corollary 2.4. Let B(x, r)be a metric ball with ∂B(x, r)=∅. If the scalar curvature
Rmr−2 inB(x, r), (2.14)
then
logvol(B(x, r))
rm inf
0<τr2μB(x,r)(g, τ)−C(m). (2.15) Proof. The proof is essentially contained in [10, Remark 13.13]. We sketch its proof here for the sake of completeness. Assumea priorithat
vol(B(x, r))4mvol
B
x,r 2
. (2.16)
Then we claim that λ4,B C(m)·r−2 under the assumption (2.14). Actually, it can be checked by choosing the test function u, in the definition of λ4,B, which is a positive constant in B(x,r2) and decreases linearly to 0 onB(x, r)\B(x,r2) such that´
Bu2= 1. Then by (2.7), we get, under (2.16), μB(x,r)(g, r2)logvol(B(x, r))
rm +C(m).
In general, there always exists a minimum integerk0 0 such that (2.16) holds for the radius 2−k0r.
Then for thisk0 we have
μB(x,2−k0r)(g,2−2k0r2)logvol(B(x,2−k0r))
2−k0mrm +C(m). (2.17) By the monotonicity ofμwith respect to the subdomains,
μB(x,r)(g,2−2k0r2)μB(x,2−k0r)(g,2−2k0r2). (2.18) On the other hand, by the definition of k0, we have by induction that
vol(B(x, r))
rm vol(B(x,2−k0r))
2−k0mrm . (2.19)
The required volume ratio estimate follows from these three formulas.
2.2 A partial monotonicity of the volume ratio under Ricci flow
We shall prove a partial monotonicity of the volume ratio under Ricci flow. It follows from a partial monotonicity of local entropy whose proof is essentially contained in [19, 22].
We first sketch the partial monotonicity of local entropy. Letr0 >0 and (M, g(t)), 0tr20,be a solution to the Ricci flow on a compact m-manifold. LetA1 be a constant and Ω =Bg(r20)(x0, Ar0) be a ball at time r20 such that ∂Ω= ∅. By approximating by smooth domains bigger than Ω and the monotonicity ofμwith respect to domains, in the following calculation we may assume that∂Ω is smooth.
Letτ0>0 be any constant and defineτ(t) =τ0+r02−t.
By Rothaus [16], there exists a minimizer of μΩ(g(r20), τ0), i.e., u ∈ C∞(Ω)∩C0(Ω) which vanishes identically on∂Ω. Soucan be viewed as a function onM which vanishes outside Ω. The Euler-Lagrange equation reads
τ0(−4Δu+Ru)−ulogu2=
μΩ(g(r02), τ0) +m
2 log(4πτ) +m
·u. (2.20)
Now letv(t) be the solution to the backward heat equation
∂
∂tv=−Δv+Rv (2.21)
with initial value v(r20) = u2(r20). By the maximal principle we have v > 0 for all 0 t < r20. Put u(t) =
v(t) whent < r20. Then we have ˆ
M
u(t)2dvg(t)= 1, (2.22)
and the Li-Yau-Perelman Harnack inequality holds (see [22, (6.3.30)] or [19, Theorem 4.2]), i.e., τ(Ru−4Δu)−ulogu2−
μΩ(g(r20), τ0) +m
2 log(4πτ) +m
·u0, ∀0t < r02. (2.23) At any time 0t < r20, we pick a cut-off functionη∈C0∞(Bg(t)(x0, r0)) such that
0η1, η≡1 onBg(t)(x0,2−1r0) (2.24) and∇ηg(t)4r−01. Letδ=ηu(t)L2(g(t)) and put ˜u=δ−1ηu(t) which satisfies
˜
u∈C0∞(Bg(t)(x0, r0)) and ˆ
M
˜
u2dvg(t)= 1.
Obviously we haveδ2´
Bg(t)(x0,r20)u2(t)dvg(t). Then, by definition and a straightforward calculation, W(g(t),u, τ˜ ) =τ
ˆ
M
(Ru˜2+ 4|∇˜u|2)dv− ˆ
M
˜
u2log ˜u2dv−m
2 log(4πτ)−m
= ˆ
M
δ−2η2[τ(Ru2−4uΔu)−u2logu2]dv+ 4τ δ−2 ˆ
M
u2|∇η|2dv
− ˆ
M
δ−2η2log(δ−2η2)·u2dv−m
2 log(4πτ)−m.
Then, by the Li-Yau-Perelman Harnack inequality (2.23), W(g(t),u, τ˜ )μΩ(g(r2), τ0) + 4τ δ−2
ˆ
M
u2|∇η|2dv− ˆ
M
δ−2η2log(δ−2η2)·u2dv.
Using the trivial fact that−xlogx1 for anyx >0, we obtain the following lemma which is essentially [19, Theorem 5.1].
Lemma 2.5. Under the above assumption we have
μB(g(t), τ(t))μΩ(g(r02), τ0) + 100·
ˆ
Bg(t)(x0,r20)
u2(t)dvg(t)
−1
(2.25) whenever0tr20 andτ0r02, where B=Bg(t)(x0, r0)andΩ =Bg(r2)(x0, Ar0).
Corollary 2.6. Let(M, g(t)),0tr20,be a Ricci flow defined on a compactm-manifold. Letx0 be a base point. Suppose that the Sobolev constant as defined in (2.9)satisfies
Cs(Bg(t)(x0, r0))C(m)·r0, ∀0tr02, (2.26) and that the scalar curvature satisfies
R(t)−mr0−2 inBg(t)(x0, r0), ∀0tr20. (2.27) Then for any metric ball Bg(r02)(x, r)⊂Bg(r20)(x0, Ar0), of the radius rr0, satisfying the scalar curva- ture bound
R(r02)mr−2 inBg(r2
0)(x, r), (2.28)
we have
logvolg(t)(Bg(t)(x0, r0))
rm0 logvolg(r2 0)(Bg(r2
0)(x, r))
rm + C(m)
inf´
Bg(t)(x0,r20)v(t)dv, (2.29) where the infimum in the last term on the right-hand side is taken over all the nonnegative conjugate heat solutions v(t)with initial valuev(r02)∈C0∞(Bg(r2
0)(x0, Ar0))satisfying ˆ
M
v(r20)dvg(r2 0)= 1.
Proof. Let Ω =Bg(r2
0)(x0, Ar0). For any 0< τ0r2r20, we have μBg(r2
0 )(x,r)(g(r02), τ0)μΩ(g(r02), τ0)μBg(t)(x0,r0)(g(t), τ(t))−´ C(m)
Bg(t)(x0,r20)v(t)dv,
where v(t) is the solution to the conjugate heat equation with initial value v(r02), the square of the minimizer of μΩ(g(r02), τ0). Then one can use the upper bound ofμBg(r2
0 )(x,r)(g(r02), τ0) in Corollary 2.4 and the lower bound of μBg(t)(x0,r0)(g(t), τ(t)) in Corollary 2.2 where τ(t) τ0+r20 2r02 to get the required volume ratio estimate.
The remaining question is how to estimate the integration´
Bg(t)(x0,r2)v(t)dvfrom below. We shall deal with this in the following section.
3 The heat kernel lower bound to the conjugate heat equation
The heat kernel estimate is one important topic in Ricci flow and many remarkable applications have been found (see [1–5, 9, 11, 14] and [20, 21, 23]). In all these works the non-collapsing assumption is essential.
In this section we establish a local Li-Yau type heat kernel lower bound which remains valid for metrics with a collapsing structure.
For simplicity in this section we assume that the space-time scale is 1. Letg(t), 0t1, be a Ricci flow on a compact m-manifold M. Let B0 = Bg(r2
0)(x0,1) be a metric ball at the initial time t = 0, centered at a point x0, such that∂B=∅. Assume that
|Ric|1 onB0×[0,1]. (3.1)
Then by the Ricci flow equation, we have
e−1·g(t)g(1)e·g(t) onB0×[0,1]. (3.2)
So
Bg(t)(x0,e−2)⊂Bg(s)(x0,e−1)⊂B0, ∀t, s∈[0,1]. (3.3) LetH(x, t;y, t), 0t< t1, be the heat kernel to the conjugate heat equation which satisfies
− ∂
∂tH = Δg(t),xH−R(x, t)·H (3.4) with initial value
tlimtH(x, t;y, t) =δg(t),y(x), (3.5)
and ∂
∂tH = Δg(t),yH (3.6)
with initial value
tlimtH(x, t;y, t) =δg(t),x(y), (3.7) whereδg(t),x is the Dirac function concentrated at (x, t) with respect to the Riemannian measure ofg(t).
For the existence of the heat kernel and its estimates under Ricci flow, please refer to [7, 22] and the references therein.
In the following we will usedtto denote the distance function ofg(t).
3.1 The heat kernel integral bound
In this subsection we adopt Li-Yau’s argument (see [13]), following the idea of Cheeger and Yau [6], to derive an integral lower bound of H. More references for the Li-Yau estimate are [12, 17].
Letϕ:R→[0,∞) be a smooth function satisfying ϕ(r) = 1, ∀r 1
2e−5k, ϕ(r) = 0, ∀re−5, ϕ0. (3.8) The function
u(y, t) = ˆ
M
H(x,0;y, t)·ϕ(d0(x0, x))dvg(0)(x) (3.9) satisfies the forward heat equation
∂
∂tu= Δg(t)u (3.10)
with initial value
tlim→0u(y, t) =ϕ(d0(x0, y)).
We define the comparison function in the Euclidean space Rmas follows:
¯
u(ζ, t) :=
ˆ
Rm
(4πt)−m/2e−|ξ|
2
4t ϕ(|ξ−ζ|)dξ. (3.11)
The function ¯uis trivially a radial function in the space factor. So ¯udetermines a function ¯u(r, t) for any r0 andt0, by simply setting ¯u(r, t) = ¯u(|ζ|, t) wheneverr=|ζ|. In the following we use ¯utand ¯ur
to denote the derivatives in the variablestand r, respectively. It is easy to check that
¯
ur(r, t)0 (3.12)
for anyr, t0. Moreover, ¯usatisfies the heat equation onRm,
¯
ut= ¯urr+n−1 r ·u¯r. It follows that
¯
utu¯rr. (3.13)
Now we define the comparison function on M,
¯
u(y, t) = ¯u(dt(x0, y), t), ∀y∈M. (3.14) It satisfies the initial condition
¯
u(y,0) =ϕ(d0(x0, y)), ∀y∈M.
Lemma 3.1. Assume (3.1). Then the functionu¯ satisfies
∂
∂tu¯Δg(t)u¯+C(m) (3.15)
in the barrier sense.
Proof. The Laplacian at timet is given by
Δ¯u= ¯ur·Δdt+ ¯urr; (3.16)
the derivation in tis given by
∂
∂tu¯= ¯ut+ ¯ur· ∂dt
∂t. (3.17)
We discuss two independent cases according to r e−5 or not. When r e−5, one can apply [14, Lemma 8.3(a)], together with the assumption (3.1), to derive ∂d∂tt Δdt−2.Thus, together with (3.12) and (3.13), the formulas (3.16) and (3.17) yield ∂t∂u¯Δ¯u+ ¯ut−u¯rr−2¯urΔ¯u−2¯ur.Now the required estimate (3.15) follows from the estimate
−u¯r(r, t) ˆ
Rm
(4πt)−m/2e−|ξ4t|2|ϕ|(|ζ−ξ|)dξC, where Cis a universal constant, andζ is any point with|ζ|=r.
When r < e−5, one can use the local Taylor expansion to get −u¯r(r, t) C(m)·r. On the other hand, since Ric −1 onBg(t)(x0,1), the Laplacian comparison gives Δdt C(m)r in the barrier sense.
Substitute the estimates into (3.16) to get Δ¯uu¯rr−C(m) in the barrier sense. Then notice that the term ∂d∂tt in (3.17) admits the estimate
∂dt
∂t (x, t)inf
− ˆ
γ
Ric( ˙γ,γ)˙
γ is a minimal geodesic connectingx0 andx
−r.
Thus,
∂
∂tu¯u¯t+C(m)u¯rr+C(m)Δ¯u+C(m) in the barrier sense.
The function ¯u(y, t) = ¯¯ u(y, t)−Ctsatisfies∂t∂u¯¯Δg(t)u¯¯and the initial condition ¯u(y,¯ 0) =ϕ(d0(x0, y)).
By the maximum principle we have
u(y, t)u(y, t),¯¯ ∀(y, t)∈M ×[0,1]. (3.18) By approximation we may choose ϕas the characteristic function on (−∞,e−4]. In particular we have the following consequence.
Corollary 3.2. Under the assumption(3.1)we have ˆ
Bg(0)(x0,e−4)
H(x,0;y, t)dvg(0)(x) ˆ
B(ζ,e−4)
(4πt)−m/2e−|ξ4|t2dξ−C(m)·t, (3.19) whereζ∈Rmis a point with|ζ|=dt(x0, y)andB(ζ,e−4)is the metric ball of the radiuse−4 centered at ζ in Rm.
In particular, whendt(x0, y)e−5the integration part on the right-hand side tends to 1 ast→0. So we have the following corollary.
Corollary 3.3. Under the assumption(3.1)we have ˆ
Bg(0)(x0,e−4)
H(x,0;y, t)dvg(0)(x) 1
2, (3.20)
wheneverdt(x0, y)e−5 and0tt0, wheret0=t0(m)1001 is a positive constant.
Remark 3.4. For the static metric case, the local curvature condition (3.1) is not sufficient to derive the heat kernel lower bound (3.20).
The same argument by replacing the time 0 byt<1 gives the following corollary.
Corollary 3.5. Under the assumption(3.1)we have ˆ
Bg(t)(x0,e−4)
H(x, t;y, t)dvg(t)(x) 1
2, (3.21)
whenever dt(x0, y)1 and0t < tt+t0, where t0=t0(m) 1001 is a positive constant depending only on the dimension m.
3.2 The Li-Yau type heat kernel lower bound
First of all we recall the Harnack inequality related to the conjugate heat kernel which was proved by Kuang and Zhang [11] (see also [22, Corollary 6.4.1]). We fix the timet= 1−t0 and any pointy∈M. Then we have
H(x2, t2;y,1−t0)H(x1, t1;y; 1−t0)·
1−t0−t1 1−t0−t2
3m/2
·exp
L(x1, t1;x2, t2) 2(t2−t1)
(3.22) for anyx1, x2 and 12 t1< t2<1−t0. Here,
L(x1, t1;x2, t2) = inf
γ
ˆ t2 t1
(t2−t1)2·R(γ(t), t) + 4 dγ
dt 2
g(t)
dt, (3.23)
where γranges over all the curves connectingx1 andx2.
Combining the integral lower bound (3.21) we can derive the pointwise Li-Yau type heat kernel lower bound.
Corollary 3.6. Under the assumption(3.1), we have
H(x,1−2t0;y,1−t0) c(m)
volg(1)(Bg(1)(x0,e−2)) (3.24) for any x∈Bg(1)(x0,e−2) and y ∈Bg(1−t0)(x0,1), where t0 = t0(m) 1001 is the positive constant in Corollary 3.5 andc(m)is a positive constant depending only on m.
Proof. Applying (3.22) witht1= 1−2t0andt2= 1−32t0 we get H
x2,1−3
2t0;y,1−t0
H(x1,1−2t0;y,1−t0)·C(m)·exp
L(x1,1−2t0;x2,1−32t0) t0
. Whenx1, x2∈Bg(1)(x0,e−2), we can choose a minimal geodesic connecting them at timet= 1, i.e., γ: [1−2t0,1−32t0]→M with constant speed2d1(xt1,x2)
0 et40. Thenγlies in the domainBg(1)(x0,e−1)⊂B0, so by the uniform equivalence of the metricsg(t) onB0 we have
L
x1,1−2t0;x2,1−3 2t0
ˆ 1−32t0 1−2t0
t20
4 ·R(γ(t), t) + 4 dγ
dt 2
g(t)
dt
ˆ 1−32t0 1−2t0
mt20 4 + 4e2
dγ dt
2
g(1)
dtC(m).
In follows that
H
x2,1−3
2t0;y,1−t0
C(m)·H(x1,1−2t0;y; 1−t0) for any x1, x2 ∈ Bg(1)(x0,e−2). Thus, using Bg(1−3
2t0)(x0,e−3) ⊂ Bg(1)(x0,e−2) we have the integral estimate
H(x1,1−2t0;y,1−t0)C(m)−1·
Bg
(1−3
2t0)(x0,e−3)
H
x,1−3
2t0;y,1−t0
dvg(1−3
2t0)(x).
Now, by (3.21) we also have ˆ
Bg(1−3
2t0)(x0,e−4)
H
x,1−3
2t0;y,1−t0
dvg(1−3
2t0)(x) 1 2 for any y ∈Bg(1−t0)(x0,e−5). SoH(x1,1−2t0;y,1−t0)C(m)−1·volg(1−3
2t0)(Bg(1−3
2t0)(x0,e−3))−1 for any y ∈ Bg(1−t0)(x0,e−5). The required heat kernel lower bound (3.24) follows from the metric equivalence (3.2) at timest= 1−32t0 andt= 1 onBg(1)(x0,1),
volg(1−3
2t0)(Bg(1−3
2t0)(x0,e−3))em·volg(1)(Bg(1−3
2t0)(x0,e−3))em·volg(1)(Bg(1)(x0,e−2)).
This completes the proof.
Next, one can follow Perelman’s argument (see [14]) to improve the heat kernel lower bound in the case wherey admits a large distance fromx0 (see also [19] for a similar argument).
As in [14], for anyA1, we letφbe a non-decreasing function of one variable, equal 1 on (−∞,12e−5), and rapidly increase to infinity on (12e−5,e−5), in such a way that
2(φ)2/φ−φ(t−01A+ 2)φ−C(m, A)·φ (3.25) for some constantC(m, A)<∞, wheret0=t0(m) is the positive constant as in Corollary 3.5. Fixx∈M and let
H(y, t) =H(x,1−2t0;y, t), 1−2t0< t1, (3.26) be a solution to the forward heat equation. One can easily check that the function
h(y, t) =H(y, t)·φ(dt(x0, y)−t−01A(t−(1−t0))) satisfies
∂
∂th= Δh−2φ−1∇φ,∇h+
φ ∂dt
∂t −Δdt−t−01A
−φ+ 2(φ)2/φ
·H.
By [14, Lemma 8.3(a)] again, we have under the assumption (3.1), ∂d∂tt −Δdt−2. Thus,
∂
∂thΔh−2φ−1∇φ,∇h −C(m, A)h. (3.27) By the maximum principle, miny∈M(eC(m,A)th(y, t)) increases int. In particular, since φ1, we have
min
Bg(t)(x0,r(t))H(·, t)min
M h(·, t)eC(m,A)·min
M h(·,1)eC(m,A)· min
Bg(1)(x0,A)H(·,1),
where r(t) = 1 +t−01A(t−(1−t0)) is a radius at timet such thathis infinite outsideBg(t)(x0, r(t)). In particular,
min
Bg(1)(x0,A)H(·,1)e−C(m,A)· min
Bg(1−t0)(x0,e−5)H(·,1−t0). (3.28)
In other words,
H(x,1−2t0;y,1)e−C(m,A)· min
z∈Bg(1−t
0)(x0,e−5)H(x,1−2t0;z,1−t0) (3.29) for anyy∈Bg(1)(x0, A). Whenx∈Bg(1)(x0,e−2), the right-hand side admits a lower bound by (3.24)
min
z∈Bg(1−t0)(x0,1)H(x,1−2t0;z,1−t0) c(m)
volg(1)(Bg(1)(x0,e−2)). So we conclude the following lemma.
Lemma 3.7. Under the assumption (3.1), we have
H(x,1−2t0;y,1) c(m)·e−C(m,A)
volg(1)(Bg(1)(x0,e−2)) (3.30) for any x ∈ Bg(1)(x0,e−2) and y ∈ Bg(1)(x0, A), where t0 = t0(m) 1001 is the positive constant in Corollary 3.5 andc(m)is a positive constant depending only on m.
Remark 3.8. According to the calculation in [19] the constant C(m, A) can be chosen as C(m)·A2 for some constantC(m).
Corollary 3.9. Let v(x, t) be a nonnegative solution to the conjugate heat equation
− ∂
∂tv= Δv−Rv (3.31)
with initial valuev∈C0(Bg(1)(x0, A))satisfying ˆ
M
v(x)dvg(1)(x) = 1. (3.32)
Then under the assumption(3.1), we have
v(x,1−2t0) c(m, A)
volg(1)(Bg(1)(x0,e−2)) (3.33) for any x∈Bg(1−2t0)(x0,e−3). In particular,
ˆ
Bg(1−2t0)(x0,e−3)
v(x,1−2t0)dvg(1−2t0)(x)c(m, A) (3.34) for some positive constant c(m, A)depending onm andA.
Proof. The solutionv has the formal representation v(x, t) =
ˆ
M
H(x, t;y,1)·v(y)dvg(1)(y) (3.35) for anyt<1. Then for anyx∈Bg(1)(x0,e−2), (3.30) yields
v(x,1−2t0) = ˆ
Bg(1)(x0,A)
H(x,1−2t0;y,1)·v(y)dvg(1)(y) c(m, A)
volg(1)(Bg(1)(x0,e−2)).
Noticing that Bg(1−2t0)(x0,e−3)⊂Bg(1)(x0,e−2) we get the estimate (3.33). The last integral estimate (3.34) is a consequence of metric equivalence ofg(1−2t0) andg(1) on the metric ballBg(1−2t0)(x0,e−2) and the relative volume comparison at time 1,
volg(1−2t0)(Bg(1−2t0)(x0,e−3))e−mvolg(1)(Bg(1−2t0)(x0,e−3)) e−mvolg(1)(Bg(1)(x0,e−4)) c(m)·volg(1)(Bg(1)(x0,e−2)).
This completes the proof.
