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Noncollapsing estimate for the Ricci-Bourguignon flow
Differential Geometry and its Applications 69 (2020) 101610
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Noncollapsing estimate for the Ricci-Bourguignon flow Ying Shen, Lin Feng Wang ∗ School of Science, Nantong University, Nantong 226007, Jiangsu, China
a r t i c l e
i n f o
Article history: Received 23 September 2019 Accepted 16 January 2020 Available online xxxx Communicated by J. Berndt MSC: 53C21 Keywords: Ricci-Bourguignon flow Sobolev inequality Noncollapsing estimate Perelman’s functional
a b s t r a c t The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow (1−(n−1)ρ)2 evolving on [0, T ), whose first eigenvalue λ0 of the operator − + 4(1−2(n−1)ρ) R for the initial metric g(0) is positive, or T > 0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to derive this noncollapsing estimate, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman’s functional, and then we can derive a Sobolev inequality along the R-B flow. © 2020 Published by Elsevier B.V.
1. Introduction Let M be an n-dimensional closed Riemannian manifold with a metric g = g(t) evolving along the Ricci-Bourguignon flow (R-B flow) ∂g = −2(Ric − ρRg), ∂t
(1.1)
where Ric is the Ricci curvature tensor, R is the scalar curvature and ρ is a real constant. This family of geometric flows contains, the Einstein flow (ρ = 12 ), the traceless Ricci flow (ρ = n1 ), the Schouten flow 1 (ρ = 2(n−1) ), and the famous Ricci flow (ρ = 0), etc [3,17]. Recall the Yamabe flow is ∂g = −(R − r)g, ∂t where r denotes the mean value of the scalar curvature [2,19]. As stated in [3], when ρ ≤ 0 the R-B flow can be seen as an interpolation between the Ricci flow and the Yamabe flow, by a suitable rescaling in time. * Corresponding author. E-mail addresses: [email protected] (Y. Shen), [email protected] (L.F. Wang). https://doi.org/10.1016/j.difgeo.2020.101610 0926-2245/© 2020 Published by Elsevier B.V.
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Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
There have been some studies on the R-B flow or its related problems. Catino-Mazzieri [4] and Dwivedi [8] studied special solutions of the R-B flow, i.e., the gradient ρ-Einstein solitons, respectively. Lu-QingZheng [14,15] considered the short time existence and the local estimate of the covariant derivative of the curvature tensor about the conformal solutions of the R-B flow. Recently, Azami [1] considered some questions of the R-B flow on the Heisenberg group and the quaternion Lie group. Catino et al. [3] found that the existence of the solution of the R-B flow depends on the constant ρ. 1 Concretely, for the case that n ≥ 3, if ρ > 2(n−1) , then the R-B flow has no short time solution for any 1 initial metric g0 ; and if ρ < 2(n−1) , then the R-B flow has a unique smooth short time solution for any initial 1 metric g0 . Hence ρ = 2(n−1) corresponds to the critical case, now the R-B flow reduces to the Schouten flow, and the short time existence of the solution has not yet been determined. The second author [18] established an evolving formula for the Perelman’s functional Wρ along the R-B flow, and then he proved that the R-B flow does not exist nontrivial breathers for some ρ. Recently, Liang-Zhu [13] proved that along the R-B flow, the boundedness of the Ricci curvature implies the boundedness of the curvature tensor, by doing local norm estimate for the Weyl tensor. Perelman [16] established two locally noncollapsing theorems for the Ricci flow (see Definition 2.1), based on the monotone formula of the functional W and the estimates of the reduced volume. Zhang [22] derived a strong noncollapsing result of the Ricci flow (see Definition 2.2), by proving a uniform Sobolev inequality, which is independent of the time of the surgery. Ye [20] established a logarithmic Sobolev inequality along the Ricci flow, based on the monotone formula of the functional W, then he derived a Sobolev inequality and got a noncollapsing estimate along the Ricci flow. Recent studies for collapsing solutions to the Ricci flow can be referred in [9–11]. In this paper we shall prove that for the R-B flow evolving on [0, T ), whose first eigenvalue λ0 of the (1−(n−1)ρ)2 operator − + 4(1−2(n−1)ρ) R for the initial metric g(0) is positive or T > 0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to do this, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman’s functional, we do this in Section 3. Similar to [7,20], we can derive the Sobolev inequality along the R-B flow from the logarithmic Sobolev inequality, we do this in Section 4. Then the strong noncollapsing estimate for solutions of the R-B flow was derived in Section 5. 2. Notations and lemmas Cheeger-Gromov firstly gave the concept of noncollapsing on the manifold [5,6]. The definitions of κ-noncollapsing or strong κ-noncollapsing along the Ricci flow can be found in [12,16,21]. Similarly, we can define the κ-noncollapsing or strong κ-noncollapsing along the R-B flow. For which, we assume that μ = n(1 − 2(n − 1)ρ) ≥ 2.
(2.1)
Definition 2.1. We use V (Bt (x0 , s), g(t)) to denote the volume of the geodesic ball Bt (x0 , s) about the metric g(t). For constants r > 0, κ > 0 and some time t0 > 0, we say that a solution of the R-B flow is √ κ-noncollapsing (on the scale r) at (x0 , t0 ), if for all s : t0 < s < r, there is V (Bt0 (x0 , s), g(t0 )) ≥ κsμ , provided the Riemannian curvature satisfies |Rie| ≤ s−2 on Bt0 (x0 , s) × [t0 − s2 , t0 ]. Moreover we say that a solution of the R-B flow on [0, T ) is κ-noncollapsing (on the scale r), if for all (x0 , t0 ) ∈ M × [0, T ), this solution is κ-noncollapsing (on the scale r) at (x0 , t0 ). Definition 2.2. For constants r > 0, κ > 0 and some time t0 > 0, we say that a solution of the R-B flow is √ strong κ-noncollapsing (on the scale r) at (x0 , t0 ), if for all s : t0 < s < r, there is
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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V (Bt0 (x0 , s), g(t0 )) ≥ κsμ , provided the scalar curvature satisfies R ≤ s−2 on Bt0 (x0 , s). Moreover we say that a solution of the R-B flow on [0, T ) is strong κ-noncollapsing (on the scale r), if for all (x0 , t0 ) ∈ M × [0, T ), this solution is strong κ-noncollapsing (on the scale r) at (x0 , t0 ). Remark 2.3. Compared with the definition of noncollapsing, the definition of strong noncollapsing needs only an upper bound of the scalar curvature on the geodesic ball Bt0 (x0 , s), and not on the parabolic ball Bt0 (x0 , s) × [t0 − s2 , t0 ]. In [18], the second author defined the Wρ -functional by Wρ (g, f, τ ) = [τ (β(ρ)R + f ) + M
where f satisfies
M
n 1 f − ln (4πτ ) − n]e−f dx, 1 − 2(n − 1)ρ 2
(2.2)
e−f dx = 1 and β(ρ) is defined by β(ρ) =
(1 − (n − 1)ρ)2 . 1 − 2(n − 1)ρ
(2.3)
Let A be a solution of n ) + A(n − 1)2 ρ]2 2 n(A + 1)2 − 2nA(n − 1)ρ). = (4(n − 1)2 ρ − 3n + 4)( 4 [(A + 1)(1 −
(2.4)
Remark 2.4. As showed in [18], (2.4) has a solution A < 0 if ρ ≤ 0. For the case that ρ ≤ 0, an evolving formula of Wρ along the R-B flow was established in [18]. Lemma 2.5. Assume that g(t) is a solution of (1.1) and ρ ≤ 0, then dWρ = 2β(ρ)τ dt
|Ric + M
+
A − 1 − 2ρτ R 2 −f 1 − (n − 1)ρ Hessf + g| e dx β(ρ) 4τ (1 − (n − 1)ρ)
1 2(1 − 2(n − 1)ρ)
(A + 1)(1 − n2 ) + A(n − 1)2 ρ 2 −f [ −τ y(ρ)ρR + ] e dx, −τ y(ρ)
(2.5)
M
here f evolves by ∂f = (nρ − 1)R − (1 − 2(n − 1)ρ)(f − |∇f |2 ), ∂t
(2.6)
A ≤ 0 is a solution of (2.4), τ > 0 evolves by dτ = A, dt
(2.7)
β(ρ) is a constant given in (2.3) and y(ρ) = 4(n − 1)2 ρ − 3n + 4.
(2.8)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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The Sobolev constant of (M, g) is defined to be CS,ρ (M, g) = sup {u
− V − n (M, g)u2 : u ∈ C 1 (M ), ∇u2 = 1}, 1
2μ μ−2
here V (M, g) is the volume of (M, g) and μ is defined in (2.1). In other words, CS,ρ (M, g) is the smallest number such that the inequality u
≤ CS,ρ (M, g)∇u2 + V − n (M, g)u2 1
2μ μ−2
(2.9)
holds true for all u ∈ W 1,2 (M ). The following two lemmas can be seen as generalizations of the logarithmic Sobolev inequalities established in [20]. Lemma 2.6. Let (M, g) be an n-dimensional closed manifold. For all a > 0 and all u ∈ W 1,2 (M ) with u2 = 1, there holds:
2 μaCS,ρ (M, g) u ln u dx ≤ 2 2
|∇u|2 dx
2
M
M 2 μ μ − ln a + [ln 2 + aV − n (M, g) − 1]. 2 2
(2.10)
Proof. By the Jensen’s inequality, we have 2μ 4 4 ln u μ−2 dx = ln u2 · u μ−2 dx ≥ u2 ln u μ−2 dx M
M
M
for u ∈ W 1,2 (M ) with u2 = 1. It follows from (2.9) that 2μ μ−2 μ ln u μ−2 dx = ln u 2μ u2 ln |u| dx ≤ μ−2 4 2 M
M 1 μ ≤ ln [CS,ρ (M, g)∇u2 + V − n (M, g)]. 2
Hence
u2 ln u2 dx ≤
1 μ ln [CS,ρ (M, g)∇u2 + V − n (M, g)]2 2
M
≤
μ μ 2 ln 2 + ln [CS,ρ (M, g) 2 2
|∇u|2 dx + V − n (M, g)]. 2
(2.11)
M
It is easy to see that for all a > 0, y ≥ 0 and x > −y, ln (x + y) ≤ ax + ay − 1 − ln a 2 is true. Choosing x = CS,ρ (M, g)
M
|∇u|2 dx and y = V − n (M, g) in above inequality leads to 2
2 ln [CS,ρ (M, g)
|∇u|2 dx + V − n (M, g)] 2
M
2 (M, g) ≤ aCS,ρ M
|∇u|2 dx + aV − n (M, g) − 1 − ln a. 2
(2.12)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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By (2.11) and (2.12) we get (2.10). 2 Lemma 2.7. Let (M, g) be an n-dimensional closed manifold. We assume that the first eigenvalue λ0 (g) of the operator − + β(ρ)R is positive. Then for all a > δ0 and all u ∈ W 1,2 (M ) with u2 = 1, there holds: 4 u2 ln u2 dx ≤
2 μaCS,ρ (M, g) 2
M
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
μ μ − ln a + ln 2 + σ0 , 2 2
(2.13)
here 2 δ0 = (λ0 (g)CS,ρ (M, g) + V − n (M, g) − 2
σ0 =
2 β(ρ)CS,ρ (M, g) min R− −1 ) , 4
μ 2 [ln δ0−1 − ln (λ0 (g)CS,ρ (M, g)) − 1], 2
(2.14) (2.15)
and R− = min {0, R}. Proof. Note that for all a > 0, y > 0, b > 0 and x ≥ b, ln (x + y) ≤ ax − ln a + ln (b + y) − ln b − 1 is true if a ≥
1 b+y
(2.16)
[20]. By (2.11) we have u2 ln u2 dx ≤
μ μ 2 ln 2 + ln [CS,ρ (M, g) 2 2
M
|∇u|2 dx + V − n (M, g)] 2
M
μ μ 2 (M, g) ≤ ln 2 + ln [CS,ρ 2 2
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
+V − n (M, g) − 2
2 (M, g) min R− β(ρ)CS,ρ
4
].
(2.17)
Choosing 2 x = CS,ρ (M, g)
(|∇u|2 +
β(ρ)R 2 u ) dx, 4
M
y = V − n (M, g) − 2
2 β(ρ)CS,ρ (M, g) min R− 4
2 and b = λ0 (g)CS,ρ (M, g) in (2.16) leads to
2 ln [CS,ρ (M, g)
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M 2 (M, g) min R− β(ρ)CS,ρ 2 ] +V − n (M, g) − 4 β(ρ)R 2 2 2 u ) dx − ln a − ln [λ0 (g)CS,ρ ≤ aCS,ρ (M, g) (|∇u|2 + (M, g)] − 1 4 M 2 (M, g) + V − n (M, g) − + ln [λ0 (g)CS,ρ 2
2 β(ρ)CS,ρ (M, g) min R− ]. 4
(2.18)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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By (2.17) and (2.18) we get (2.13). 2 3. Logarithmic Sobolev inequality along the R-B flow For convenience we define the Wρ∗ -functional by Wρ∗ (g, f, τ ) =
[τ (β(ρ)R + f ) + M
1 f ]e−f dx. 1 − 2(n − 1)ρ
Due to (2.2), we have Wρ (g, f, τ ) = Wρ∗ (g, f, τ ) −
n ln (4πτ ) − n. 2
We define μ∗ρ (g, τ ) = inf {Wρ∗ (g, f, τ ) : f ∈ C∞ (M ),
e−f dx = 1}.
(3.1)
M
Theorem 2.5 shows that
dWρ dt
≥ 0, which implies that dWρ∗ (g, f, τ ) n d ≥ ln τ, dt 2 dt
(3.2)
here g = g(t) is a smooth solution of the R-B flow on M × [0, T ) for some (finite or infinite) T > 0, f (t), τ (t) satisfy (2.6) and (2.7), respectively. Let 0 < t∗ < T and σ > 0. Note that A < 0, hence we set T ∗ = t∗ + σ ∗ and τ = τ (t) = T ∗ + At for 0 ≤ t ≤ − tA . There is a minimizer ft2 , such that μ∗ρ (g(t2 ), τ (t2 )) = Wρ∗ (g(t2 ), ft2 , τ (t2 )). We can solve equation (2.6) backward in [t1 , t2 ] with initial value f (t2 ) = ft2 , and we will get a solution f (t1 ) with M e−f (t1 ) dx = 1. So Wρ∗ (g(t1 ), f (t1 ), τ (t1 )) +
n τ (t2 ) ln ≤ Wρ∗ (g(t2 ), ft2 , τ (t2 )) = μ∗ρ (g(t2 ), τ (t2 )), 2 τ (t1 )
which implies that μ∗ρ (g(t1 ), τ (t1 )) ≤ Wρ∗ (g(t1 ), f (t1 ), τ (t1 )) ≤ μ∗ρ (g(t2 ), τ (t2 )) +
n τ (t1 ) ln . 2 τ (t2 )
∗
Choosing t1 = 0 and t2 = − tA , we then arrive at μ∗ρ (g(0), t∗ + σ) ≤ μ∗ρ (g(−
n t∗ + σ t∗ ), σ) + ln . A 2 σ
Since 0 < t∗ < T is arbitrary, we conclude that n t+σ t μ∗ρ (g(0), t + σ) ≤ μ∗ρ (g(− ), σ) + ln A 2 σ
(3.3)
holds for all t ∈ [0, T ) and σ > 0 (the case t = 0 is trivial). By using the ideas given in [20] and the inequality (3.3), we can derive the following logarithmic Sobolev inequality along the R-B flow.
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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Theorem 3.1. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). Then for each σ > 0 and each t ∈ [0, T ) there holds u2 ln u2 dx M
≤σ
(
β(ρ)R 2 nσ − 4Aμt u + |∇u|2 ) dx − β(ρ) min R t=0 4 4n
M
σ nσ − 4Aμt μ , + 1) + − (ln 2 2 2 ˜ ˜ 2 μCS,ρ (M, g(0)) nCS,ρ (M, g(0))V n (M, g(0))
(3.4)
here μ is given in (2.1), and C˜S,ρ (M, g) = max{CS,ρ (M, g), 1} is the modified Sobolev constant. Proof. Let e−f = u2 . It is easy to verify that Wρ∗ can be rewritten by
Wρ∗ =
[τ (β(ρ)Ru2 + 4|∇u|2 ) − M
1 u2 ln u2 ] dx. 1 − 2(n − 1)ρ
(3.5)
By (2.10) we have u2 ln u2 dx ≤
μaC˜S,ρ (M, g) 2
M
|∇u|2 dx M
2 μ μ − ln a + [ln 2 + aV − n (M, g) − 1]. 2 2
(3.6)
We choose g = g(0) and a=
8(t + σ) 8(t + σ)(1 − 2(n − 1)ρ) = 2 (M, g(0)) 2 (M, g(0)) μC˜S,ρ nC˜S,ρ
in (3.6) and deduce 1 1 − 2(n − 1)ρ
u2 ln u2 dx ≤ 4(t + σ)
M
|∇u|2 dx − M
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
8(t + σ) n − 1). + (ln 2 + 2 2 2 nC˜S,ρ (M, g(0))V n (M, g(0)) By (3.5) we have W ∗ (g(0), u, t + σ) ≥ (t + σ)β(ρ)
Ru2 dx + M
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
8(t + σ) n − 1) − (ln 2 + 2 2 (M, g(0))V n 2 nC˜S,ρ (M, g(0))
(3.7)
8
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
≥ (t + σ)β(ρ) min R + t=0
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
n 8(t + σ) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) which implies that μ∗ (g(0), t + σ) ≥ (t + σ)β(ρ) min R + t=0
8(t + σ) n ln 2 (M, g(0)) 2 nC˜S,ρ
8(t + σ) n − (ln 2 + − 1). 2 2 (M, g(0))V n 2 nC˜S,ρ (M, g(0))
(3.8)
By (3.3) and (3.8) we have n t+σ 8(t + σ) t n ≥ (t + σ)β(ρ) min R + + ln μ∗ρ (g(− ), σ) + ln 2 ˜ t=0 A 2 σ 2 nCS,ρ (M, g(0)) n 8(t + σ) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) or μ∗ρ (g(t), σ) ≥ (σ − At)β(ρ) min R + t=0
8σ n ln 2 (M, g(0)) 2 nC˜S,ρ
n 8(σ − At) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) which implies that 1 1 − 2(n − 1)ρ
u2 ln u2 dx M
(β(ρ)Ru2 + 4|∇u|2 ) dx − (σ − At)β(ρ) min R
≤σ
t=0
M
−
4σ n 4(σ − At) n ln − , + 2 2 2 ˜ ˜ 2 nCS,ρ (M, g(0)) CS,ρ (M, g(0))V n (M, g(0)) 2
which is equivalent to (3.4). 2 The logarithmic Sobolev inequality in Theorem 3.1 depends on the time, especially when time becomes large. The next result takes care of large time under the assumption that a certain eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. 4 Theorem 3.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then for all t ∈ [0, T ), σ > 0 satisfying 4 σ − 4(1 − 2(n − 1)ρ)At ≥ and all u ∈ W 1,2 (M ) with u2 = 1, there holds:
μ 2 C (M, g(0))δ0 (g(0)), 2 S,ρ
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
u ln u dx − σ 2
2
M
(
9
β(ρ) 2 Ru + |∇u|2 ) dx 4
M
σ μ , ≤ σ0 − ln 2 2 μCS,ρ (M, g(0))
(3.9)
here δ0 , σ0 are defined in (2.14) and (2.15) respectively, with g = g(0). Proof. Assume that μ C 2 (M, g(0))δ0 . 8(1 − 2(n − 1)ρ) S,ρ
t+σ ≥ We set a=
8(t + σ) 8(1 − 2(n − 1)ρ)(t + σ) = . 2 2 μCS,ρ (M, g(0)) nCS,ρ (M, g(0))
Then a ≥ δ0 . Using this a in (2.13) we deduce that for all u ∈ W 1,2 (M ) with u2 = 1,
u ln u dx ≤ 4(1 − 2(n − 1)ρ)(t + σ) 2
2
M
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
4(t + σ) μ + σ0 , − ln 2 2 nCS,ρ (M, g(0)) which implies that μ∗ (g(0), t + σ) ≥
4(t + σ) σ0 n ln − . 2 2 nCS,ρ (M, g(0)) 1 − 2(n − 1)ρ
(3.10)
By (3.3) and (3.10) we have n t+σ n 4(t + σ) σ0 t ≥ ln μ∗ρ (g(− ), σ) + ln 2 (M, g(0)) − 1 − 2(n − 1)ρ , A 2 σ 2 nCS,ρ or μ∗ρ (g(t), σ) ≥
4σ σ0 n ln 2 (M, g(0)) − 1 − 2(n − 1)ρ , 2 nCS,ρ
here σ − At ≥
μ C 2 (M, g(0))δ0 . 8(1 − 2(n − 1)ρ) S,ρ
We then get that [σ(β(ρ)Ru2 + 4|∇u|2 ) − M
≥
1 u2 ln u2 ] dx 1 − 2(n − 1)ρ
4σ σ0 n ln − . 2 2 nCS,ρ (M, g(0)) 1 − 2(n − 1)ρ
Replacing 4(1 − 2(n − 1)ρ)σ in (3.11) by σ leads to (3.9). 2
(3.11)
10
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
Based on Theorem 3.1 and Theorem 3.2, we can prove the following logarithmic Sobolev inequality along the R-B flow. Theorem 3.3. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then for all t ∈ [0, T ), σ > 0, there holds: 4
u ln u dx ≤ σ 2
2
M
(
β(ρ) 2 μ Ru + |∇u|2 ) dx − ln σ + C 4 2
(3.12)
M
holds for all u ∈ W 1,2 (M ) with u2 = 1, where β(ρ) is defined in (2.3), and C depends only on n, ρ and the geometric quantities of the initial metric. Proof. For all t ∈ [0, T ) and σ > 0, if σ − 4(1 − 2(n − 1)ρ)At ≤
μ 2 C (M, g(0))δ0 , 2 S,ρ
then (3.12) comes from Theorem 3.1 directly; otherwise (3.12) comes from Theorem 3.2 directly. 2 4. Sobolev inequality along the R-B flow Let Ψ ∈ L∞ (M ). The quadratic form corresponding to the potential function Ψ is defined by (|∇u|2 + Ψu2 ) dx.
Q(u) =
(4.1)
M
The following lemma is established in [20], which shows that how to derive the Sobolev inequality from the logarithmic Sobolev inequality. Lemma 4.1. Let 0 < σ∗ < +∞. Assume that for each 0 < σ < σ ∗ the logarithmic Sobolev inequality u2 ln u2 dx ≤ σQ(u) −
ln σ + C 2
(4.2)
M
holds for all u ∈ W 1,2 (M ) and u2 = 1, where , C are constants such that > 2. Then we have the Sobolev inequality u22 ≤ ( −2
− σ ∗ 1− n ¯ )(Q(u) + 4 − σ ∗ min Ψ u2 ) ) C(C, 2 ∗ 4 σ
(4.3)
¯ ) can be bounded from above in for all u ∈ W 1,2 (M ), where Ψ− = min {0, Ψ}, the positive constant C(C, 1 ¯ terms of upper bounds for C, and −2 and −n n− 3σ ∗ 1 min Ψ− + C]. C¯ = 2 2 (σ ∗ ) 4 exp [ − 4 16 2
In this section we prove the Sobolev inequality along the R-B flow from the logarithmic Sobolev inequality in Theorem 3.3.
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
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Theorem 4.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then there exists a constant B, depending only on n, ρ 4 and the geometric quantities of the initial metric, such that for all t ∈ [0, T ) and u ∈ W 1,2 (M ), u22μ ≤ B
(
μ−2
β(ρ) 2 Ru + |∇u|2 ) dx, 4
(4.4)
M
here all geometric quantities except B are associated with g(t). Proof. Theorem 3.3 tells us that u2 ln u2 dx ≤ σQ(u) −
μ ln σ + C1 , 2
(4.5)
M
here C1 depends only on n, ρ and the geometric quantities of the initial metric, and Q(u) =
(
β(ρ) 2 Ru + |∇u|2 ) dx. 4
(4.6)
M
Choosing σ ∗ = 4, Ψ =
β(ρ) 4 R
in Lemma 4.1, we have
u22μ ≤ C2 [ μ−2
(
β(ρ) 2 β(ρ) Ru + |∇u|2 ) dx + (1 − min R− ) t 4 4
M
u2 dx], M
here C2 = C2 (n, ρ, − mint R− ) is a positive constant depending on n, ρ, − mint R− . As pointed out in [3], ∂t R ≥ (1 − 2(n − 1)ρ)R since ρ ≤ 0. Then the maximum principle shows that the minimum of the scalar curvature is nondecreasing along the R-B flow, i.e., mint R− ≥ mint=0 R− . We also know from [18] that the first eigenvalue λ0 (g(t)) of the operator − + β(ρ)R is nondecreasing along the R-B flow. Hence 4 u22μ μ−2 β(ρ) 2 β(ρ) 2 − Ru + |∇u| ) dx + (1 − min R ) u2 dx] ≤ C2 [ ( 4 4 t=0 M
≤ C2 (1 +
1 β(ρ) min R− }) max {0, 1 − λ0 4 t=0
M
(
β(ρ) 2 Ru + |∇u|2 ) dx. 4
M
We then arrive at (4.4). 2 Similar to Theorem 4.2, we have the following Sobolev inequality along the R-B flow. Now the condition λ0 (g(0)) > 0 is not assumed, but the bounds also depend on an upper bound of time. Theorem 4.3. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) with a given initial value g(0). We assume that T > 0 is finite. Then there exist constants B, D, depending only on n, ρ, the geometric quantities of the initial metric and the upper bound of T , such that for all t ∈ [0, T ) and u ∈ W 1,2 (M ),
12
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
u
≤B
2
2μ μ−2
β(ρ) 2 Ru + |∇u|2 ) dx + D ( 4
M
u2 dx,
(4.7)
M
here all geometric quantities except B, D are associated with g(t). Proof. Theorem 3.1 tells us that u2 ln u2 dx ≤ σQ(u) −
μ ln σ + C3 , 2
M
here Q(u) is given in (4.6), and C3 = − + ≤− +
1 nσ − 4Aμt μ β(ρ) min R − (ln + 1) 2 (M, g(0)) t=0 4n 2 μC˜S,ρ nσ − 4Aμt 2 2 ˜ nCS,ρ (M, g(0))V n (M, g(0)) nσ − 4AμT 1 μ β(ρ) min R− − (ln + 1) 2 ˜ t=0 4n 2 μCS,ρ (M, g(0)) nσ − 4AμT . 2 2 ˜ nCS,ρ (M, g(0))V n (M, g(0))
The rest of the proof is similar as in Theorem 4.2. 2 5. Strong noncollpasing estimate In this section we will show that the R-B flow is strong noncollapsed relative to upper bounds of the scalar curvature on all scales (see Definition 2.2), if the first eigenvalue λ0 of the operator − + β(ρ)R for 4 the initial metric g(0) is positive, or the time is finite. For which we firstly give the following lemma. Theorem 5.1. Let (M, g) be an n-dimensional closed manifold. We assume that for B, D > 0 the Sobolev inequality u22μ ≤ B
(
μ−2
β(ρ) 2 Ru + |∇u|2 ) dx + D 4
M
holds for all u ∈ W 1,2 (M ). Assume that R ≤ V (B(x0 , r)) ≥ (
u2 dx
(5.1)
M 1 r2
on a geodesic ball B(x0 , r) with r > 0. Then 4
(β(ρ)B + 4
μ 2 +2
μ
B+
4Dr2 )
) 2 rμ .
(5.2)
Proof. For convenience, we use V (Ω) to denote the volume of Ω ⊂ M for the metric g(t). Assume that R ≤ r12 on a geodesic ball B(x0 , r), but the estimate (5.2) does not hold, i.e. V (B(x0 , r)) < (
4 (β(ρ)B + 4
μ 2 +2
μ
B+
4Dr2 )
) 2 rμ .
(5.3)
We will derive a contradiction. Due to (5.1), for u ∈ C ∞ (M ) with support contained in B(x0 , r) we have
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
|u|
(
2μ μ−2
dx)
μ−2 μ
≤B
B(x0 ,r)
≤B
β(ρ) 2 Ru + |∇u|2 ) dx + D ( 4
B(x0 ,r)
13
u2 dx
B(x0 ,r)
β(ρ)B |∇u| dx + ( + D) 4r2
2
B(x0 ,r)
u2 dx.
(5.4)
B(x0 ,r)
By the Hölder’s inequality, we have
u dx ≤ ( B(x0 ,r)
2μ
|u| μ−2 dx)
2
μ−2 μ
2
V μ (B(x0 , r)).
(5.5)
B(x0 ,r)
Plugging (5.5) into (5.4) and using (5.3), we have
μ
2μ
|u| μ−2 dx)
(
μ−2 μ
(β(ρ) + 4 2 +2 )B + 4Dr2 μ 4 2 +2
≤
B(x0 ,r)
|∇u|2 dx.
(5.6)
B(x0 ,r)
Next consider an arbitrary domain Ω ⊂ B(x0 , r). For u ∈ C ∞ (M ) with support contained in Ω, by (5.5) and (5.6) we have
μ
u2 dx ≤
(β(ρ) + 4 2 +2 )B + 4Dr2 μ2 V (Ω) μ 4 2 +2
Ω
|∇u|2 dx, Ω
which implies that μ
4 2 +2 λ1 (Ω)V (Ω) ≥ , μ (β(ρ) + 4 2 +2 )B + 4Dr2 2 μ
(5.7)
here λ1 (Ω) denotes the first nonzero Dirichlet eigenvalue of on Ω. For B(x0 , s) ⊂ B(x0 , r), let u(y) = s − d(x0 , y), then λ1 (B(x0 , s))
≤ ≤
B(x0 ,s)
|∇u|2 dx
B(x0 ,s)
u2 dx
B(x0 ,s)
≤
|∇u|2 dx
B(x0 , s2 )
u2 dx
4 V (B(x0 , s)) . · s2 V (B(x0 , 2s ))
(5.8)
By (5.7) and (5.8) we have μ
V (B(x0 , s)) ≥ (
μ μ 4 2 +2 s2 ) 2+μ 4− 2+μ V μ +2 2 (β(ρ) + 4 2 )B + 4Dr
μ 2+μ
s (B(x0 , )). 2
Iterating (5.9) we obtain V (B(x0 , s)) μ
m m μ μ μ l l m s 4 2 +2 s2 l=1 ( 2+μ ) 4− l=1 l( 2+μ ) V ( 2+μ ) (B(x , ) )) ≥( μ 0 m +2 2 2 2 (β(ρ) + 4 )B + 4Dr μ
∞ ∞ μ μ l l 4 2 +2 s2 l=1 ( 2+μ ) 4− l=1 l( 2+μ ) ) →( μ +2 2 2 (β(ρ) + 4 )B + 4Dr
(5.9)
14
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
as m → ∞, here we have used the fact that μ
m
lim V ( 2+μ ) (B(x0 ,
m→∞
s )) = 1. 2m
Hence μ
V (B(x0 , s)) ≥ (
μ(μ+2) μ 4 2 +2 s2 ) 2 4− 4 . μ +2 2 (β(ρ) + 4 2 )B + 4Dr
In particular, we have V (B(x0 , r)) ≥ (
4 (β(ρ) + 4
μ 2 +2
μ
)B +
4Dr2
) 2 rμ ,
which contradicts to (5.3). Then we finish the proof. 2 When the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive, we have the 4 following strong noncollapsing estimate for the R-B flow, which can be seen as a corollary of Theorem 4.2 and Theorem 5.1. Theorem 5.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Assume that R ≤ r12 on a geodesic ball B(x0 , r) 4 with r > 0. Then for all t ∈ [0, T ), V (B(x0 , r), g(t)) ≥ (
μ 4 ) 2 rμ , μ +2 (β(ρ) + 4 2 )B
(5.10)
here B is from Theorem 4.2. A similar strong noncollapsing estimate for bounded time follows from Theorem 4.3 and Theorem 5.1, for which the condition λ0 > 0 is not assumed. Theorem 5.3. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) with a given initial value g(0). We assume that T > 0 is finite. Assume that R ≤ r12 on a geodesic ball B(x0 , r) with r > 0. Then for all t ∈ [0, T ), V (B(x0 , r), g(t)) ≥ (
μ 4 ) 2 rμ , μ (β(ρ)B + 4 2 +2 B + 4Dr2 )
(5.11)
here B, D are from Theorem 4.3. References [1] S. Azami, The Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups, arXiv:1810.04283v1 [math.DG]. [2] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differ. Geom. 69 (2) (2005) 217–278. [3] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, L. Mazzieri, The Ricci-Bourguignon flow, Pac. J. Math. 287 (2) (2017) 337–370. [4] G. Catino, L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132 (2016) 66–94. [5] J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded; I, J. Differ. Geom. 23 (1986) 309–346. [6] J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded; II, J. Differ. Geom. 32 (1990) 269–298. [7] E.B. Davies, Heat Kernel and Spectral Theory, Cambridge University Press, 1989.
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[8] S. Dwivedi, Some results on Ricci-Bourguignon and Ricci-Bourguignon almost solitons, arXiv:1809.11103v1 [math.DG]. [9] S. Gindi, J. Streets, Structure of collapsing solutions of generalized Ricci flow, arXiv:1811.08877v1 [math.DG]. [10] H.Z. Huang, L.L. Kong, X.C. Rong, S.C. Xu, Collapsed manifolds with Ricci bounded covering geometry, arXiv:1808. 03774v1 [math.DG]. [11] S. Huang, Notes on Ricci flows with collapsing time-slices (I): distance distortion, arXiv:1808.07394v2 [math.DG]. [12] B. Kleiner, J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (5) (2008) 2587–2855. [13] Q.T. Liang, A.Q. Zhu, On the conditions to extend Ricci Bourguignon flow, Nonlinear Anal. 179 (2019) 344–353. [14] P. Lu, J. Qing, Y. Zheng, A note on the conformal Ricci flow, Pac. J. Math. 268 (2014) 413–434. [15] P. Lu, J. Qing, Y. Zheng, Conformal Ricci flow on asymptotically hyperbolic manifolds, Sci. China Math. 62 (1) (2019) 157–170. [16] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159v1. [17] S. Pigola, M. Rigoli, M. Rimoldi, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 10 (4) (2010) 757–799. [18] L.F. Wang, Monotonicity of eigenvalues and functionals along the Ricci-Bourguignon flow, J. Geom. Anal. 29 (2019) 1116–1135. [19] R. Ye, Global existence and convergence of Yamabe flow, J. Differ. Geom. 39 (1) (1994) 35–50. [20] R. Ye, The logarithmic Sobolev inequality along the Ricci flow, arXiv:0707.2424v2 [math.DG]. [21] R. Ye, Curvature estimates for the Ricci flow; I, Calc. Var. Partial Differ. Equ. 31 (4) (2008) 417–437. [22] Q.S. Zhang, Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries, Pac. J. Math. 239 (1) (2007) 179–200.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Noncollapsing estimate for the Ricci-Bourguignon flow Ying Shen, Lin Feng Wang ∗ School of Science, Nantong University, Nantong 226007, Jiangsu, China
a r t i c l e
i n f o
Article history: Received 23 September 2019 Accepted 16 January 2020 Available online xxxx Communicated by J. Berndt MSC: 53C21 Keywords: Ricci-Bourguignon flow Sobolev inequality Noncollapsing estimate Perelman’s functional
a b s t r a c t The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow (1−(n−1)ρ)2 evolving on [0, T ), whose first eigenvalue λ0 of the operator − + 4(1−2(n−1)ρ) R for the initial metric g(0) is positive, or T > 0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to derive this noncollapsing estimate, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman’s functional, and then we can derive a Sobolev inequality along the R-B flow. © 2020 Published by Elsevier B.V.
1. Introduction Let M be an n-dimensional closed Riemannian manifold with a metric g = g(t) evolving along the Ricci-Bourguignon flow (R-B flow) ∂g = −2(Ric − ρRg), ∂t
(1.1)
where Ric is the Ricci curvature tensor, R is the scalar curvature and ρ is a real constant. This family of geometric flows contains, the Einstein flow (ρ = 12 ), the traceless Ricci flow (ρ = n1 ), the Schouten flow 1 (ρ = 2(n−1) ), and the famous Ricci flow (ρ = 0), etc [3,17]. Recall the Yamabe flow is ∂g = −(R − r)g, ∂t where r denotes the mean value of the scalar curvature [2,19]. As stated in [3], when ρ ≤ 0 the R-B flow can be seen as an interpolation between the Ricci flow and the Yamabe flow, by a suitable rescaling in time. * Corresponding author. E-mail addresses: [email protected] (Y. Shen), [email protected] (L.F. Wang). https://doi.org/10.1016/j.difgeo.2020.101610 0926-2245/© 2020 Published by Elsevier B.V.
2
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
There have been some studies on the R-B flow or its related problems. Catino-Mazzieri [4] and Dwivedi [8] studied special solutions of the R-B flow, i.e., the gradient ρ-Einstein solitons, respectively. Lu-QingZheng [14,15] considered the short time existence and the local estimate of the covariant derivative of the curvature tensor about the conformal solutions of the R-B flow. Recently, Azami [1] considered some questions of the R-B flow on the Heisenberg group and the quaternion Lie group. Catino et al. [3] found that the existence of the solution of the R-B flow depends on the constant ρ. 1 Concretely, for the case that n ≥ 3, if ρ > 2(n−1) , then the R-B flow has no short time solution for any 1 initial metric g0 ; and if ρ < 2(n−1) , then the R-B flow has a unique smooth short time solution for any initial 1 metric g0 . Hence ρ = 2(n−1) corresponds to the critical case, now the R-B flow reduces to the Schouten flow, and the short time existence of the solution has not yet been determined. The second author [18] established an evolving formula for the Perelman’s functional Wρ along the R-B flow, and then he proved that the R-B flow does not exist nontrivial breathers for some ρ. Recently, Liang-Zhu [13] proved that along the R-B flow, the boundedness of the Ricci curvature implies the boundedness of the curvature tensor, by doing local norm estimate for the Weyl tensor. Perelman [16] established two locally noncollapsing theorems for the Ricci flow (see Definition 2.1), based on the monotone formula of the functional W and the estimates of the reduced volume. Zhang [22] derived a strong noncollapsing result of the Ricci flow (see Definition 2.2), by proving a uniform Sobolev inequality, which is independent of the time of the surgery. Ye [20] established a logarithmic Sobolev inequality along the Ricci flow, based on the monotone formula of the functional W, then he derived a Sobolev inequality and got a noncollapsing estimate along the Ricci flow. Recent studies for collapsing solutions to the Ricci flow can be referred in [9–11]. In this paper we shall prove that for the R-B flow evolving on [0, T ), whose first eigenvalue λ0 of the (1−(n−1)ρ)2 operator − + 4(1−2(n−1)ρ) R for the initial metric g(0) is positive or T > 0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to do this, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman’s functional, we do this in Section 3. Similar to [7,20], we can derive the Sobolev inequality along the R-B flow from the logarithmic Sobolev inequality, we do this in Section 4. Then the strong noncollapsing estimate for solutions of the R-B flow was derived in Section 5. 2. Notations and lemmas Cheeger-Gromov firstly gave the concept of noncollapsing on the manifold [5,6]. The definitions of κ-noncollapsing or strong κ-noncollapsing along the Ricci flow can be found in [12,16,21]. Similarly, we can define the κ-noncollapsing or strong κ-noncollapsing along the R-B flow. For which, we assume that μ = n(1 − 2(n − 1)ρ) ≥ 2.
(2.1)
Definition 2.1. We use V (Bt (x0 , s), g(t)) to denote the volume of the geodesic ball Bt (x0 , s) about the metric g(t). For constants r > 0, κ > 0 and some time t0 > 0, we say that a solution of the R-B flow is √ κ-noncollapsing (on the scale r) at (x0 , t0 ), if for all s : t0 < s < r, there is V (Bt0 (x0 , s), g(t0 )) ≥ κsμ , provided the Riemannian curvature satisfies |Rie| ≤ s−2 on Bt0 (x0 , s) × [t0 − s2 , t0 ]. Moreover we say that a solution of the R-B flow on [0, T ) is κ-noncollapsing (on the scale r), if for all (x0 , t0 ) ∈ M × [0, T ), this solution is κ-noncollapsing (on the scale r) at (x0 , t0 ). Definition 2.2. For constants r > 0, κ > 0 and some time t0 > 0, we say that a solution of the R-B flow is √ strong κ-noncollapsing (on the scale r) at (x0 , t0 ), if for all s : t0 < s < r, there is
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
3
V (Bt0 (x0 , s), g(t0 )) ≥ κsμ , provided the scalar curvature satisfies R ≤ s−2 on Bt0 (x0 , s). Moreover we say that a solution of the R-B flow on [0, T ) is strong κ-noncollapsing (on the scale r), if for all (x0 , t0 ) ∈ M × [0, T ), this solution is strong κ-noncollapsing (on the scale r) at (x0 , t0 ). Remark 2.3. Compared with the definition of noncollapsing, the definition of strong noncollapsing needs only an upper bound of the scalar curvature on the geodesic ball Bt0 (x0 , s), and not on the parabolic ball Bt0 (x0 , s) × [t0 − s2 , t0 ]. In [18], the second author defined the Wρ -functional by Wρ (g, f, τ ) = [τ (β(ρ)R + f ) + M
where f satisfies
M
n 1 f − ln (4πτ ) − n]e−f dx, 1 − 2(n − 1)ρ 2
(2.2)
e−f dx = 1 and β(ρ) is defined by β(ρ) =
(1 − (n − 1)ρ)2 . 1 − 2(n − 1)ρ
(2.3)
Let A be a solution of n ) + A(n − 1)2 ρ]2 2 n(A + 1)2 − 2nA(n − 1)ρ). = (4(n − 1)2 ρ − 3n + 4)( 4 [(A + 1)(1 −
(2.4)
Remark 2.4. As showed in [18], (2.4) has a solution A < 0 if ρ ≤ 0. For the case that ρ ≤ 0, an evolving formula of Wρ along the R-B flow was established in [18]. Lemma 2.5. Assume that g(t) is a solution of (1.1) and ρ ≤ 0, then dWρ = 2β(ρ)τ dt
|Ric + M
+
A − 1 − 2ρτ R 2 −f 1 − (n − 1)ρ Hessf + g| e dx β(ρ) 4τ (1 − (n − 1)ρ)
1 2(1 − 2(n − 1)ρ)
(A + 1)(1 − n2 ) + A(n − 1)2 ρ 2 −f [ −τ y(ρ)ρR + ] e dx, −τ y(ρ)
(2.5)
M
here f evolves by ∂f = (nρ − 1)R − (1 − 2(n − 1)ρ)(f − |∇f |2 ), ∂t
(2.6)
A ≤ 0 is a solution of (2.4), τ > 0 evolves by dτ = A, dt
(2.7)
β(ρ) is a constant given in (2.3) and y(ρ) = 4(n − 1)2 ρ − 3n + 4.
(2.8)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
4
The Sobolev constant of (M, g) is defined to be CS,ρ (M, g) = sup {u
− V − n (M, g)u2 : u ∈ C 1 (M ), ∇u2 = 1}, 1
2μ μ−2
here V (M, g) is the volume of (M, g) and μ is defined in (2.1). In other words, CS,ρ (M, g) is the smallest number such that the inequality u
≤ CS,ρ (M, g)∇u2 + V − n (M, g)u2 1
2μ μ−2
(2.9)
holds true for all u ∈ W 1,2 (M ). The following two lemmas can be seen as generalizations of the logarithmic Sobolev inequalities established in [20]. Lemma 2.6. Let (M, g) be an n-dimensional closed manifold. For all a > 0 and all u ∈ W 1,2 (M ) with u2 = 1, there holds:
2 μaCS,ρ (M, g) u ln u dx ≤ 2 2
|∇u|2 dx
2
M
M 2 μ μ − ln a + [ln 2 + aV − n (M, g) − 1]. 2 2
(2.10)
Proof. By the Jensen’s inequality, we have 2μ 4 4 ln u μ−2 dx = ln u2 · u μ−2 dx ≥ u2 ln u μ−2 dx M
M
M
for u ∈ W 1,2 (M ) with u2 = 1. It follows from (2.9) that 2μ μ−2 μ ln u μ−2 dx = ln u 2μ u2 ln |u| dx ≤ μ−2 4 2 M
M 1 μ ≤ ln [CS,ρ (M, g)∇u2 + V − n (M, g)]. 2
Hence
u2 ln u2 dx ≤
1 μ ln [CS,ρ (M, g)∇u2 + V − n (M, g)]2 2
M
≤
μ μ 2 ln 2 + ln [CS,ρ (M, g) 2 2
|∇u|2 dx + V − n (M, g)]. 2
(2.11)
M
It is easy to see that for all a > 0, y ≥ 0 and x > −y, ln (x + y) ≤ ax + ay − 1 − ln a 2 is true. Choosing x = CS,ρ (M, g)
M
|∇u|2 dx and y = V − n (M, g) in above inequality leads to 2
2 ln [CS,ρ (M, g)
|∇u|2 dx + V − n (M, g)] 2
M
2 (M, g) ≤ aCS,ρ M
|∇u|2 dx + aV − n (M, g) − 1 − ln a. 2
(2.12)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
5
By (2.11) and (2.12) we get (2.10). 2 Lemma 2.7. Let (M, g) be an n-dimensional closed manifold. We assume that the first eigenvalue λ0 (g) of the operator − + β(ρ)R is positive. Then for all a > δ0 and all u ∈ W 1,2 (M ) with u2 = 1, there holds: 4 u2 ln u2 dx ≤
2 μaCS,ρ (M, g) 2
M
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
μ μ − ln a + ln 2 + σ0 , 2 2
(2.13)
here 2 δ0 = (λ0 (g)CS,ρ (M, g) + V − n (M, g) − 2
σ0 =
2 β(ρ)CS,ρ (M, g) min R− −1 ) , 4
μ 2 [ln δ0−1 − ln (λ0 (g)CS,ρ (M, g)) − 1], 2
(2.14) (2.15)
and R− = min {0, R}. Proof. Note that for all a > 0, y > 0, b > 0 and x ≥ b, ln (x + y) ≤ ax − ln a + ln (b + y) − ln b − 1 is true if a ≥
1 b+y
(2.16)
[20]. By (2.11) we have u2 ln u2 dx ≤
μ μ 2 ln 2 + ln [CS,ρ (M, g) 2 2
M
|∇u|2 dx + V − n (M, g)] 2
M
μ μ 2 (M, g) ≤ ln 2 + ln [CS,ρ 2 2
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
+V − n (M, g) − 2
2 (M, g) min R− β(ρ)CS,ρ
4
].
(2.17)
Choosing 2 x = CS,ρ (M, g)
(|∇u|2 +
β(ρ)R 2 u ) dx, 4
M
y = V − n (M, g) − 2
2 β(ρ)CS,ρ (M, g) min R− 4
2 and b = λ0 (g)CS,ρ (M, g) in (2.16) leads to
2 ln [CS,ρ (M, g)
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M 2 (M, g) min R− β(ρ)CS,ρ 2 ] +V − n (M, g) − 4 β(ρ)R 2 2 2 u ) dx − ln a − ln [λ0 (g)CS,ρ ≤ aCS,ρ (M, g) (|∇u|2 + (M, g)] − 1 4 M 2 (M, g) + V − n (M, g) − + ln [λ0 (g)CS,ρ 2
2 β(ρ)CS,ρ (M, g) min R− ]. 4
(2.18)
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
6
By (2.17) and (2.18) we get (2.13). 2 3. Logarithmic Sobolev inequality along the R-B flow For convenience we define the Wρ∗ -functional by Wρ∗ (g, f, τ ) =
[τ (β(ρ)R + f ) + M
1 f ]e−f dx. 1 − 2(n − 1)ρ
Due to (2.2), we have Wρ (g, f, τ ) = Wρ∗ (g, f, τ ) −
n ln (4πτ ) − n. 2
We define μ∗ρ (g, τ ) = inf {Wρ∗ (g, f, τ ) : f ∈ C∞ (M ),
e−f dx = 1}.
(3.1)
M
Theorem 2.5 shows that
dWρ dt
≥ 0, which implies that dWρ∗ (g, f, τ ) n d ≥ ln τ, dt 2 dt
(3.2)
here g = g(t) is a smooth solution of the R-B flow on M × [0, T ) for some (finite or infinite) T > 0, f (t), τ (t) satisfy (2.6) and (2.7), respectively. Let 0 < t∗ < T and σ > 0. Note that A < 0, hence we set T ∗ = t∗ + σ ∗ and τ = τ (t) = T ∗ + At for 0 ≤ t ≤ − tA . There is a minimizer ft2 , such that μ∗ρ (g(t2 ), τ (t2 )) = Wρ∗ (g(t2 ), ft2 , τ (t2 )). We can solve equation (2.6) backward in [t1 , t2 ] with initial value f (t2 ) = ft2 , and we will get a solution f (t1 ) with M e−f (t1 ) dx = 1. So Wρ∗ (g(t1 ), f (t1 ), τ (t1 )) +
n τ (t2 ) ln ≤ Wρ∗ (g(t2 ), ft2 , τ (t2 )) = μ∗ρ (g(t2 ), τ (t2 )), 2 τ (t1 )
which implies that μ∗ρ (g(t1 ), τ (t1 )) ≤ Wρ∗ (g(t1 ), f (t1 ), τ (t1 )) ≤ μ∗ρ (g(t2 ), τ (t2 )) +
n τ (t1 ) ln . 2 τ (t2 )
∗
Choosing t1 = 0 and t2 = − tA , we then arrive at μ∗ρ (g(0), t∗ + σ) ≤ μ∗ρ (g(−
n t∗ + σ t∗ ), σ) + ln . A 2 σ
Since 0 < t∗ < T is arbitrary, we conclude that n t+σ t μ∗ρ (g(0), t + σ) ≤ μ∗ρ (g(− ), σ) + ln A 2 σ
(3.3)
holds for all t ∈ [0, T ) and σ > 0 (the case t = 0 is trivial). By using the ideas given in [20] and the inequality (3.3), we can derive the following logarithmic Sobolev inequality along the R-B flow.
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
7
Theorem 3.1. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). Then for each σ > 0 and each t ∈ [0, T ) there holds u2 ln u2 dx M
≤σ
(
β(ρ)R 2 nσ − 4Aμt u + |∇u|2 ) dx − β(ρ) min R t=0 4 4n
M
σ nσ − 4Aμt μ , + 1) + − (ln 2 2 2 ˜ ˜ 2 μCS,ρ (M, g(0)) nCS,ρ (M, g(0))V n (M, g(0))
(3.4)
here μ is given in (2.1), and C˜S,ρ (M, g) = max{CS,ρ (M, g), 1} is the modified Sobolev constant. Proof. Let e−f = u2 . It is easy to verify that Wρ∗ can be rewritten by
Wρ∗ =
[τ (β(ρ)Ru2 + 4|∇u|2 ) − M
1 u2 ln u2 ] dx. 1 − 2(n − 1)ρ
(3.5)
By (2.10) we have u2 ln u2 dx ≤
μaC˜S,ρ (M, g) 2
M
|∇u|2 dx M
2 μ μ − ln a + [ln 2 + aV − n (M, g) − 1]. 2 2
(3.6)
We choose g = g(0) and a=
8(t + σ) 8(t + σ)(1 − 2(n − 1)ρ) = 2 (M, g(0)) 2 (M, g(0)) μC˜S,ρ nC˜S,ρ
in (3.6) and deduce 1 1 − 2(n − 1)ρ
u2 ln u2 dx ≤ 4(t + σ)
M
|∇u|2 dx − M
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
8(t + σ) n − 1). + (ln 2 + 2 2 2 nC˜S,ρ (M, g(0))V n (M, g(0)) By (3.5) we have W ∗ (g(0), u, t + σ) ≥ (t + σ)β(ρ)
Ru2 dx + M
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
8(t + σ) n − 1) − (ln 2 + 2 2 (M, g(0))V n 2 nC˜S,ρ (M, g(0))
(3.7)
8
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
≥ (t + σ)β(ρ) min R + t=0
8(t + σ) n ln 2 ˜ 2 nCS,ρ (M, g(0))
n 8(t + σ) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) which implies that μ∗ (g(0), t + σ) ≥ (t + σ)β(ρ) min R + t=0
8(t + σ) n ln 2 (M, g(0)) 2 nC˜S,ρ
8(t + σ) n − (ln 2 + − 1). 2 2 (M, g(0))V n 2 nC˜S,ρ (M, g(0))
(3.8)
By (3.3) and (3.8) we have n t+σ 8(t + σ) t n ≥ (t + σ)β(ρ) min R + + ln μ∗ρ (g(− ), σ) + ln 2 ˜ t=0 A 2 σ 2 nCS,ρ (M, g(0)) n 8(t + σ) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) or μ∗ρ (g(t), σ) ≥ (σ − At)β(ρ) min R + t=0
8σ n ln 2 (M, g(0)) 2 nC˜S,ρ
n 8(σ − At) − (ln 2 + − 1), 2 2 ˜ 2 nCS,ρ (M, g(0))V n (M, g(0)) which implies that 1 1 − 2(n − 1)ρ
u2 ln u2 dx M
(β(ρ)Ru2 + 4|∇u|2 ) dx − (σ − At)β(ρ) min R
≤σ
t=0
M
−
4σ n 4(σ − At) n ln − , + 2 2 2 ˜ ˜ 2 nCS,ρ (M, g(0)) CS,ρ (M, g(0))V n (M, g(0)) 2
which is equivalent to (3.4). 2 The logarithmic Sobolev inequality in Theorem 3.1 depends on the time, especially when time becomes large. The next result takes care of large time under the assumption that a certain eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. 4 Theorem 3.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then for all t ∈ [0, T ), σ > 0 satisfying 4 σ − 4(1 − 2(n − 1)ρ)At ≥ and all u ∈ W 1,2 (M ) with u2 = 1, there holds:
μ 2 C (M, g(0))δ0 (g(0)), 2 S,ρ
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
u ln u dx − σ 2
2
M
(
9
β(ρ) 2 Ru + |∇u|2 ) dx 4
M
σ μ , ≤ σ0 − ln 2 2 μCS,ρ (M, g(0))
(3.9)
here δ0 , σ0 are defined in (2.14) and (2.15) respectively, with g = g(0). Proof. Assume that μ C 2 (M, g(0))δ0 . 8(1 − 2(n − 1)ρ) S,ρ
t+σ ≥ We set a=
8(t + σ) 8(1 − 2(n − 1)ρ)(t + σ) = . 2 2 μCS,ρ (M, g(0)) nCS,ρ (M, g(0))
Then a ≥ δ0 . Using this a in (2.13) we deduce that for all u ∈ W 1,2 (M ) with u2 = 1,
u ln u dx ≤ 4(1 − 2(n − 1)ρ)(t + σ) 2
2
M
(|∇u|2 +
β(ρ)R 2 u ) dx 4
M
4(t + σ) μ + σ0 , − ln 2 2 nCS,ρ (M, g(0)) which implies that μ∗ (g(0), t + σ) ≥
4(t + σ) σ0 n ln − . 2 2 nCS,ρ (M, g(0)) 1 − 2(n − 1)ρ
(3.10)
By (3.3) and (3.10) we have n t+σ n 4(t + σ) σ0 t ≥ ln μ∗ρ (g(− ), σ) + ln 2 (M, g(0)) − 1 − 2(n − 1)ρ , A 2 σ 2 nCS,ρ or μ∗ρ (g(t), σ) ≥
4σ σ0 n ln 2 (M, g(0)) − 1 − 2(n − 1)ρ , 2 nCS,ρ
here σ − At ≥
μ C 2 (M, g(0))δ0 . 8(1 − 2(n − 1)ρ) S,ρ
We then get that [σ(β(ρ)Ru2 + 4|∇u|2 ) − M
≥
1 u2 ln u2 ] dx 1 − 2(n − 1)ρ
4σ σ0 n ln − . 2 2 nCS,ρ (M, g(0)) 1 − 2(n − 1)ρ
Replacing 4(1 − 2(n − 1)ρ)σ in (3.11) by σ leads to (3.9). 2
(3.11)
10
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
Based on Theorem 3.1 and Theorem 3.2, we can prove the following logarithmic Sobolev inequality along the R-B flow. Theorem 3.3. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then for all t ∈ [0, T ), σ > 0, there holds: 4
u ln u dx ≤ σ 2
2
M
(
β(ρ) 2 μ Ru + |∇u|2 ) dx − ln σ + C 4 2
(3.12)
M
holds for all u ∈ W 1,2 (M ) with u2 = 1, where β(ρ) is defined in (2.3), and C depends only on n, ρ and the geometric quantities of the initial metric. Proof. For all t ∈ [0, T ) and σ > 0, if σ − 4(1 − 2(n − 1)ρ)At ≤
μ 2 C (M, g(0))δ0 , 2 S,ρ
then (3.12) comes from Theorem 3.1 directly; otherwise (3.12) comes from Theorem 3.2 directly. 2 4. Sobolev inequality along the R-B flow Let Ψ ∈ L∞ (M ). The quadratic form corresponding to the potential function Ψ is defined by (|∇u|2 + Ψu2 ) dx.
Q(u) =
(4.1)
M
The following lemma is established in [20], which shows that how to derive the Sobolev inequality from the logarithmic Sobolev inequality. Lemma 4.1. Let 0 < σ∗ < +∞. Assume that for each 0 < σ < σ ∗ the logarithmic Sobolev inequality u2 ln u2 dx ≤ σQ(u) −
ln σ + C 2
(4.2)
M
holds for all u ∈ W 1,2 (M ) and u2 = 1, where , C are constants such that > 2. Then we have the Sobolev inequality u22 ≤ ( −2
− σ ∗ 1− n ¯ )(Q(u) + 4 − σ ∗ min Ψ u2 ) ) C(C, 2 ∗ 4 σ
(4.3)
¯ ) can be bounded from above in for all u ∈ W 1,2 (M ), where Ψ− = min {0, Ψ}, the positive constant C(C, 1 ¯ terms of upper bounds for C, and −2 and −n n− 3σ ∗ 1 min Ψ− + C]. C¯ = 2 2 (σ ∗ ) 4 exp [ − 4 16 2
In this section we prove the Sobolev inequality along the R-B flow from the logarithmic Sobolev inequality in Theorem 3.3.
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
11
Theorem 4.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Then there exists a constant B, depending only on n, ρ 4 and the geometric quantities of the initial metric, such that for all t ∈ [0, T ) and u ∈ W 1,2 (M ), u22μ ≤ B
(
μ−2
β(ρ) 2 Ru + |∇u|2 ) dx, 4
(4.4)
M
here all geometric quantities except B are associated with g(t). Proof. Theorem 3.3 tells us that u2 ln u2 dx ≤ σQ(u) −
μ ln σ + C1 , 2
(4.5)
M
here C1 depends only on n, ρ and the geometric quantities of the initial metric, and Q(u) =
(
β(ρ) 2 Ru + |∇u|2 ) dx. 4
(4.6)
M
Choosing σ ∗ = 4, Ψ =
β(ρ) 4 R
in Lemma 4.1, we have
u22μ ≤ C2 [ μ−2
(
β(ρ) 2 β(ρ) Ru + |∇u|2 ) dx + (1 − min R− ) t 4 4
M
u2 dx], M
here C2 = C2 (n, ρ, − mint R− ) is a positive constant depending on n, ρ, − mint R− . As pointed out in [3], ∂t R ≥ (1 − 2(n − 1)ρ)R since ρ ≤ 0. Then the maximum principle shows that the minimum of the scalar curvature is nondecreasing along the R-B flow, i.e., mint R− ≥ mint=0 R− . We also know from [18] that the first eigenvalue λ0 (g(t)) of the operator − + β(ρ)R is nondecreasing along the R-B flow. Hence 4 u22μ μ−2 β(ρ) 2 β(ρ) 2 − Ru + |∇u| ) dx + (1 − min R ) u2 dx] ≤ C2 [ ( 4 4 t=0 M
≤ C2 (1 +
1 β(ρ) min R− }) max {0, 1 − λ0 4 t=0
M
(
β(ρ) 2 Ru + |∇u|2 ) dx. 4
M
We then arrive at (4.4). 2 Similar to Theorem 4.2, we have the following Sobolev inequality along the R-B flow. Now the condition λ0 (g(0)) > 0 is not assumed, but the bounds also depend on an upper bound of time. Theorem 4.3. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) with a given initial value g(0). We assume that T > 0 is finite. Then there exist constants B, D, depending only on n, ρ, the geometric quantities of the initial metric and the upper bound of T , such that for all t ∈ [0, T ) and u ∈ W 1,2 (M ),
12
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
u
≤B
2
2μ μ−2
β(ρ) 2 Ru + |∇u|2 ) dx + D ( 4
M
u2 dx,
(4.7)
M
here all geometric quantities except B, D are associated with g(t). Proof. Theorem 3.1 tells us that u2 ln u2 dx ≤ σQ(u) −
μ ln σ + C3 , 2
M
here Q(u) is given in (4.6), and C3 = − + ≤− +
1 nσ − 4Aμt μ β(ρ) min R − (ln + 1) 2 (M, g(0)) t=0 4n 2 μC˜S,ρ nσ − 4Aμt 2 2 ˜ nCS,ρ (M, g(0))V n (M, g(0)) nσ − 4AμT 1 μ β(ρ) min R− − (ln + 1) 2 ˜ t=0 4n 2 μCS,ρ (M, g(0)) nσ − 4AμT . 2 2 ˜ nCS,ρ (M, g(0))V n (M, g(0))
The rest of the proof is similar as in Theorem 4.2. 2 5. Strong noncollpasing estimate In this section we will show that the R-B flow is strong noncollapsed relative to upper bounds of the scalar curvature on all scales (see Definition 2.2), if the first eigenvalue λ0 of the operator − + β(ρ)R for 4 the initial metric g(0) is positive, or the time is finite. For which we firstly give the following lemma. Theorem 5.1. Let (M, g) be an n-dimensional closed manifold. We assume that for B, D > 0 the Sobolev inequality u22μ ≤ B
(
μ−2
β(ρ) 2 Ru + |∇u|2 ) dx + D 4
M
holds for all u ∈ W 1,2 (M ). Assume that R ≤ V (B(x0 , r)) ≥ (
u2 dx
(5.1)
M 1 r2
on a geodesic ball B(x0 , r) with r > 0. Then 4
(β(ρ)B + 4
μ 2 +2
μ
B+
4Dr2 )
) 2 rμ .
(5.2)
Proof. For convenience, we use V (Ω) to denote the volume of Ω ⊂ M for the metric g(t). Assume that R ≤ r12 on a geodesic ball B(x0 , r), but the estimate (5.2) does not hold, i.e. V (B(x0 , r)) < (
4 (β(ρ)B + 4
μ 2 +2
μ
B+
4Dr2 )
) 2 rμ .
(5.3)
We will derive a contradiction. Due to (5.1), for u ∈ C ∞ (M ) with support contained in B(x0 , r) we have
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
|u|
(
2μ μ−2
dx)
μ−2 μ
≤B
B(x0 ,r)
≤B
β(ρ) 2 Ru + |∇u|2 ) dx + D ( 4
B(x0 ,r)
13
u2 dx
B(x0 ,r)
β(ρ)B |∇u| dx + ( + D) 4r2
2
B(x0 ,r)
u2 dx.
(5.4)
B(x0 ,r)
By the Hölder’s inequality, we have
u dx ≤ ( B(x0 ,r)
2μ
|u| μ−2 dx)
2
μ−2 μ
2
V μ (B(x0 , r)).
(5.5)
B(x0 ,r)
Plugging (5.5) into (5.4) and using (5.3), we have
μ
2μ
|u| μ−2 dx)
(
μ−2 μ
(β(ρ) + 4 2 +2 )B + 4Dr2 μ 4 2 +2
≤
B(x0 ,r)
|∇u|2 dx.
(5.6)
B(x0 ,r)
Next consider an arbitrary domain Ω ⊂ B(x0 , r). For u ∈ C ∞ (M ) with support contained in Ω, by (5.5) and (5.6) we have
μ
u2 dx ≤
(β(ρ) + 4 2 +2 )B + 4Dr2 μ2 V (Ω) μ 4 2 +2
Ω
|∇u|2 dx, Ω
which implies that μ
4 2 +2 λ1 (Ω)V (Ω) ≥ , μ (β(ρ) + 4 2 +2 )B + 4Dr2 2 μ
(5.7)
here λ1 (Ω) denotes the first nonzero Dirichlet eigenvalue of on Ω. For B(x0 , s) ⊂ B(x0 , r), let u(y) = s − d(x0 , y), then λ1 (B(x0 , s))
≤ ≤
B(x0 ,s)
|∇u|2 dx
B(x0 ,s)
u2 dx
B(x0 ,s)
≤
|∇u|2 dx
B(x0 , s2 )
u2 dx
4 V (B(x0 , s)) . · s2 V (B(x0 , 2s ))
(5.8)
By (5.7) and (5.8) we have μ
V (B(x0 , s)) ≥ (
μ μ 4 2 +2 s2 ) 2+μ 4− 2+μ V μ +2 2 (β(ρ) + 4 2 )B + 4Dr
μ 2+μ
s (B(x0 , )). 2
Iterating (5.9) we obtain V (B(x0 , s)) μ
m m μ μ μ l l m s 4 2 +2 s2 l=1 ( 2+μ ) 4− l=1 l( 2+μ ) V ( 2+μ ) (B(x , ) )) ≥( μ 0 m +2 2 2 2 (β(ρ) + 4 )B + 4Dr μ
∞ ∞ μ μ l l 4 2 +2 s2 l=1 ( 2+μ ) 4− l=1 l( 2+μ ) ) →( μ +2 2 2 (β(ρ) + 4 )B + 4Dr
(5.9)
14
Y. Shen, L.F. Wang / Differential Geometry and its Applications 69 (2020) 101610
as m → ∞, here we have used the fact that μ
m
lim V ( 2+μ ) (B(x0 ,
m→∞
s )) = 1. 2m
Hence μ
V (B(x0 , s)) ≥ (
μ(μ+2) μ 4 2 +2 s2 ) 2 4− 4 . μ +2 2 (β(ρ) + 4 2 )B + 4Dr
In particular, we have V (B(x0 , r)) ≥ (
4 (β(ρ) + 4
μ 2 +2
μ
)B +
4Dr2
) 2 rμ ,
which contradicts to (5.3). Then we finish the proof. 2 When the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive, we have the 4 following strong noncollapsing estimate for the R-B flow, which can be seen as a corollary of Theorem 4.2 and Theorem 5.1. Theorem 5.2. Let (M, g) be an n-dimensional closed manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) for T > 0 with a given initial value g(0). We assume that the first eigenvalue λ0 of the operator − + β(ρ)R for the initial metric g(0) is positive. Assume that R ≤ r12 on a geodesic ball B(x0 , r) 4 with r > 0. Then for all t ∈ [0, T ), V (B(x0 , r), g(t)) ≥ (
μ 4 ) 2 rμ , μ +2 (β(ρ) + 4 2 )B
(5.10)
here B is from Theorem 4.2. A similar strong noncollapsing estimate for bounded time follows from Theorem 4.3 and Theorem 5.1, for which the condition λ0 > 0 is not assumed. Theorem 5.3. Let M be an n-dimensional closed Riemannian manifold, whose metric g = g(t) evolves along the R-B flow on [0, T ) with a given initial value g(0). We assume that T > 0 is finite. Assume that R ≤ r12 on a geodesic ball B(x0 , r) with r > 0. Then for all t ∈ [0, T ), V (B(x0 , r), g(t)) ≥ (
μ 4 ) 2 rμ , μ (β(ρ)B + 4 2 +2 B + 4Dr2 )
(5.11)
here B, D are from Theorem 4.3. References [1] S. Azami, The Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups, arXiv:1810.04283v1 [math.DG]. [2] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differ. Geom. 69 (2) (2005) 217–278. [3] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, L. Mazzieri, The Ricci-Bourguignon flow, Pac. J. Math. 287 (2) (2017) 337–370. [4] G. Catino, L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132 (2016) 66–94. [5] J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded; I, J. Differ. Geom. 23 (1986) 309–346. [6] J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded; II, J. Differ. Geom. 32 (1990) 269–298. [7] E.B. Davies, Heat Kernel and Spectral Theory, Cambridge University Press, 1989.
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