An upper bound of the heat kernel along the harmonic-Ricci flow
aa r X i v : . [ m a t h . DG ] F e b AN UPPER BOUND OF THE HEAT KERNEL ALONG THEHARMONIC-RICCI FLOW
SHOUWEN FANG AND TAO ZHENG
Abstract.
In this paper, we first derive a Sobolev inequality along the harmonic-Ricciflow. We then prove a linear parabolic estimate based on the Sobolev inequality andMoser’s iteration. As an application, we will obtain an upper bound estimate for theheat kernel under the flow. Introduction
Let M n be an n dimensional closed smooth manifold and assume n ≥
3. In [19], M¨ullerstudied a system of the Ricci flow coupled with a harmonic map heat flow (cid:26) ∂ t g = − Ric + 2 α ( t ) ∇ φ ⊗ ∇ φ,∂ t φ = τ g φ, (1.1)where φ ( · , t ) : ( M, g ( · , t )) → ( N, h ) is a family of smooth maps between two Riemannianmanifolds, both g ( · , t ) and h are Riemannian metrics, α ( t ) is a positive non-increasingfunction, and τ g φ denotes the intrinsic Laplacian of φ . This flow is also called harmonic-Ricci flow (cf. [2, 19, 27]). The harmonic-Ricci flow may be one of helpful tools in findingthe harmonic map between two Riemannian manifolds. If the target manifold N is R ,the harmonic-Ricci flow reduces to the extended Ricci flow, which was first introduced byList in [17]. The extended Ricci flow is very useful in general relativity. If φ is a constantmap, the system (1.1) degenerates to Hamilton’s Ricci flow discussed widely recently, seefor example the book [5] and seminal papers [4, 10, 11, 20]. Similarly as Ricci flow and theextended Ricci flow, corresponding theories for the harmonic-Ricci flow were establishedin [19], such as the short time existence, the W entropy, the F entropy, reduced lengthand reduced volume. Hence the harmonic-Ricci flow may be investigated through themethods used in the Ricci flow.In this paper, along the harmonic-Ricci flow, we consider the heat kernel G ( x, t, y, s ) whichis the fundamental solution of the following heat equation(∆ − ∂ t ) u ( x, t ) = 0 . (1.2)The estimate for heat kernel has always been an interesting topic in the study of differentialequations on manifolds. In their celebrated paper [16], Li and Yau derived some point-wise gradient estimates for the positive solutions of (1.2) on complete manifolds with fixedmetric and lower bounded Ricci curvature, from which the upper and lower bounds onthe heat kernel were obtained. Wang [21] proved a global gradient estimate when the Mathematics Subject Classification.
Key words and phrases. heat kernel, harmonic-Ricci flow, Sobolev inequality, log-Sobolev inequality. oundary of manifold is nonconvex, and got both upper and lower bounds for the heatkernel with Neumann conditions. Later, in [1, 2, 18] the evolved metrics were studied,and some bounds on heat kernel under some geometric flows (e.g. the Ricci flow and theextended Ricci flow) were also derived with the assistance of the Sobolev inequality.It is well-known that the Sobolev inequality is an important analytical tool in geometricanalysis. Recently, there occur many interesting results on the Sobolev inequality underdifferent geometric flows, especially the Ricci flow. In [12, 14, 22, 23, 24, 25, 26], someuniform Sobolev inequalities were proved along Ricci flow by using the monotonicity ofPerelman’s W entropy. As a consequence, Perelman’s short time non-collapsing was ex-tended to a long time version. In particular, by the Sobolev inequality, Zhang [24] proveda global upper bound for the fundamental solution of a heat equation under backwardRicci flow with the assumption that Ricci curvature is nonnegative and the injectivityradius is bounded from below.The main purpose of this paper is to establish the uniform Sobolev inequality and anupper bound for the heat kernel under the harmonic-Ricci flow. For convenience, wedenote as in [6, 7, 17, 19] the symmetric two-tensor field S y with components S ij and itstrace S := g ij S ij by S ij := R ij − α ( t ) ∇ i φ ∇ j φ and S := R − α ( t ) |∇ φ | , where R ij and R are the Ricci curvature components and the scalar curvature of ( M, g )respectively. Using the monotonicity of the W entropy, we obtain the following Sobolevinequality. Theorem 1.1.
Let ( g ( x, t ) , φ ( x, t ) ) be a solution to the harmonic-Ricci flow (1.1) in M n × [0 , T ) with initial value ( g , φ ). Let A and B be positive numbers such that thefollowing L Sobolev inequality holds initially, i.e. for any v ∈ W , ( M, g ) , (cid:18)Z M v nn − d µ ( g ) (cid:19) n − n ≤ A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ (cid:0) g (cid:1) + B Z M v d µ ( g ) . Then for all v ∈ W , ( M, g ( t )) we have (cid:18)Z M v nn − d µ (cid:0) g ( t ) (cid:1)(cid:19) n − n ≤ A ( t ) Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ (cid:0) g ( t ) (cid:1) + B ( t ) Z M v d µ (cid:0) g ( t ) (cid:1) , where A ( t ) = Ce tB nA A , B ( t ) = Ce tB nA B , and C is a positive constant depending onlyon A , B , g , φ and n . In particular, if S ≥ at the initial time, then C is a positiveconstant depending only on n . Based on the Sobolev inequality and Moser’s iteration, Ye [22] proved a linear parabolicestimate under the Ricci flow, which was applied to get the upper bound of curvaturetensor. Jiang [13] gave a linear parabolic estimate along the K¨ahler-Ricci flow, fromwhich he obtained upper bound estimates of the scalar curvature and the gradient ofRicci potential. Here from the above Sobolev inequality, we can get the following linearparabolic estimate. heorem 1.2. Assume that ( g ( x, t ) , φ ( x, t ) ) is a smooth solution to the harmonic-Ricciflow (1.1) in M n × [0 , T ] with initial value ( g , φ ) and S ≥ at the initial time. Let f be a nonnegative Lipschitz continuous function on M × [0 , T ] satisfying ∂ t f ≤ ∆ f + af (1.3) on M × [0 , T ] in the weak sense, where a ≥ . Then we have for any < t ≤ T and p > x ∈ M | f ( x, t ) | ≤ (cid:18) C a + C t (cid:19) n +22 p (cid:18)Z T Z M f p d µ d t (cid:19) p , where C and C are both positive constants depending on dimension n , p , g , φ and thefirst eigenvalue λ of F entropy with respect to g . Obviously, the heat equation (1.2) is a simple linear parabolic equation and the heat kernelsatisfies naturally the conditions of the above theorem. As a consequence of Theorem 1.2,it is not difficult to get an upper bound of the heat kernel under the harmonic-Ricci flow,which is similar to the upper bound in Bailesteanu [1], Bailesteanu and Tran [2], andWang [21]. But our upper bound depends on the first eigenvalue of F entropy, which isdifferent from their results. More precisely, we prove Theorem 1.3.
Assume that ( g ( x, t ) , φ ( x, t ) ) is a smooth solution to the harmonic-Ricciflow (1.1) in M n × [0 , T ] with initial value ( g , φ ) and S ≥ at the initial time. Let G ( x, t ; y, s ) be the heat kernel. Then there exists a positive constant C , which depends ondimension n , g , φ and the first eigenvalue λ of F entropy with respect to g , such that G ( x, t ; y, s ) ≤ C ( t − s ) n , for ∀ ≤ s < t ≤ T , and ∀ x, y ∈ M .Remark . Here the nonnegativity of S in our theorem is a little weaker than thepositivity in Bailesteanu and Tran [2]. Indeed, the assumption can be taken away if weimpose other reliance of constant C on the upper bound of time (see Corollary 3.3).The rest of the paper is organized as follows. In section 2 we consider the W entropy underthe harmonic-Ricci flow and derive the Sobolev inequality by using the monotonicity of W entropy. In section 3 we show the linear parabolic estimate for (1.3) and the upperbound estimate of the heat kernel along the harmonic-Ricci flow.2. Sobolev inequalities under the harmonic-Ricci flow
In this section, we show a uniform log-Sobolev inequality along the harmonic-Ricci flowfrom the monotonicity of W entropy, and then verify the equivalence of our Sobolevinequality and the uniform log-Sobolev inequality, from which we prove Theorem 1.1.As a corollary, we obtain the other uniform Sobolev inequality depending on the firsteigenvalue of F entropy, which will be used in the next section. irst let us introduce the definition of W entropy via corresponding conjugate heat equa-tion just as Perelman [20] has done in Ricci flow. Let u ( x, t ) be a positive solution to thefollowing conjugate heat equation ∆ u − Su + ∂ t u = 0 . (2.1)From the conjugate heat equation (2.1) and the harmonic-Ricci flow (1.1), we can geteasily dd t Z M u ( x, t )d µ ( g ( t )) = Z M ( ∂ t − S ) u d µ ( g ( t )) = − Z M ∆ u d µ ( g ( t )) = 0 , where we used the fact that M is closed. Therefore, without loss of generality we assumethat u ( x, t ) satisfies Z M u ( x, t )d µ ( g ( t )) = 1for any t ∈ [0 , T ]. The W entropy is given by a functional of the positive solution u of(2.1) as follows. Definition 2.1.
The W entropy is defined as the following functional W ( g, f, τ ) = Z M (cid:0) τ ( S + |∇ f | ) + f − n (cid:1) u d µ ( g ( t )) , where f = − ln u − n (ln 4 πτ ) and τ is a scaling factor satisfied d τ d t = − Remark . The same definition can also be found in [17, 19]. From the relationshipbetween f and u , W entropy can also be rewritten by the function u directly as follow. W ( g, u, τ ) := Z M (cid:20) τ (cid:18) Su + |∇ u | u (cid:19) − u ln u − n πτ ) u − nu (cid:21) d µ ( g ( t )) . (2.2)Now let us recall the following monotonicity formula, which had been proved in Theorem5.2 of [9] for general geometric flow and Proposition 7.1 of [19](or Theorem 6.1 of [17])for the case of constant α . We omit the details here. Proposition 2.1.
Let ( g ( x, t ) , φ ( x, t ) ) be a solution of the harmonic-Ricci flow (1.1) and u ( x, t ) be a positive solution of (2.1). Then W entropy is non-decreasing in t . Moreprecisely, dd t W = Z M (cid:16) τ | S y + Hess ( f ) − g τ | + 2 τ α | τ g φ − h∇ φ, ∇ f i| − τ ˙ α |∇ φ | (cid:17) u d µ ( g ( t )) ≥ . To prove Theorem 1.1, we first need to prove the corresponding log-Sobolev inequalityfor any t ∈ [0 , T ). Here we use the same method as Zhang [25] and Liu-Wang [18]. Usingthe monotonicity of W entropy, we have the following log-Sobolev inequality. Lemma 2.2 (Log-Sobolev Inequality) . Under the same assumptions of Theorem 1.1.Then for any t ∈ [0 , T ) , v ∈ W , ( M, g ( t )) with R M v d µ ( g ( t )) = 1 and any ǫ > , wehave Z M v ln v d µ ( g ( t )) ≤ ǫ Z M (cid:0) |∇ v | + Sv (cid:1) d µ ( g ( t )) − n ln(2 ǫ ) + 4( t + ǫ ) B A + n nA e . roof. For any fixed t ∈ [0 , T ) and any ǫ >
0, we set τ ( t ) = ǫ + t − t. From themonotonicity of the W entropy in Proposition 2.1, we getinf R M u d µ ( g )=1 W ( g , f , t + ǫ ) ≤W ( g , e f ( · , , t + ǫ ) ≤W ( g ( t ) , e f ( · , t ) , ǫ )= inf R M u d µ ( g ( t ))=1 W ( g ( t ) , f, ǫ ) , (2.3)where (4 πτ ) − n e − e f ( · ,t ) satisfies the conjugate heat equation (2.1), f and f are given bythe formulas u = (cid:0) π ( t + ǫ ) (cid:1) − n e − f and u = (4 πǫ ) − n e − f . The last equality holdsbecause the infimum of W entropy is achieved by a minimizer e f ( · , t ) (cf. Corollary 1.5.9in [4] or section 3 in [20]). Using (2.2) we rewrite (2.3) asinf R u d µ ( g )=1 Z M h ( ǫ + t )( S + |∇ ln u | ) − ln u − n (cid:0) π ( t + ǫ ) (cid:1)i u d µ ( g ) ≤ inf R u d µ ( g ( t ))=1 Z M h ǫ (cid:0) S + |∇ ln u | (cid:1) − ln u − n (cid:0) πǫ (cid:1)i u d µ ( g ( t )) . Let v = √ u and v = √ u , the above inequality leads toinf R v d µ ( g )=1 Z M (cid:2) ( ǫ + t )( Sv + 4 |∇ v | ) − v ln v (cid:3) d µ ( g ) − n t + ǫ ) ≤ inf R v d µ ( g ( t ))=1 Z M (cid:2) ǫ ( Sv + 4 |∇ v | ) − v ln v (cid:3) d µ ( g ( t )) − n ǫ . (2.4)Notice that ln x is a concave function and R v d µ ( g ) = 1, thus applying Jensen’s inequal-ity we deduce Z M v ln v q − d µ ( g ) ≤ ln Z v q − v d µ ( g ) , where q = nn − . This means Z M v ln v d µ ( g ) ≤ n k v k q , By the assumption that the Sobolev inequality holds for the initial time t = 0, combiningwith the above inequality we have Z M v ln v d µ ( g ) ≤ n (cid:18) A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ ( g ) + B (cid:19) . Moreover, the inequality ln z ≤ yz − ln y − y, z >
0. Using it in the RHSof the above we arrive at Z M v ln v d µ ( g ) ≤ n y (cid:18) A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ ( g ) + B (cid:19) − n y − n . ow we choose y = t + ǫ ) nA , then the above inequality implies Z M v ln v d µ ( g ) ≤ ( t + ǫ ) Z M (cid:0) |∇ v | + Sv (cid:1) d µ ( g ) + 4( t + ǫ ) B A − n t + ǫ ) nA − n . (2.5)Substituting (2.5) into (2.4), we conclude that Z M v ln v d µ ( g ( t )) ≤ ǫ Z M (cid:0) |∇ v | + Sv (cid:1) d µ ( g ( t )) − n ln(2 ǫ )+ 4( t + ǫ ) B A + n nA e . The time t is arbitrary, thus the proof of the lemma is completed now. (cid:3) In general, the log-Sobolev inequality and the Sobolev inequality are equivalent, whichcan be proved via the upper bound of heat kernel. More details can be found in Zhang’sTheorem 4.2.1 of [26]. But the Sobolev inequality along a geometric flow is differentfrom the general Sobolev inequality in closed manifolds. So it is necessary to provide theequivalence between our log-Sobolev inequality and Sobolev inequality here. We can givea proof of the following equivalence lemma by the trick in [3].
Lemma 2.3.
Let ( M n , g ) be a closed Riemannian manifold ( n ≥ ). Then the followinginequalities are equivalent up to constants.(I) Sobolev inequality: there exist positive constants A and B such that, for all v ∈ W , ( M ) , (cid:18)Z M v nn − d µ (cid:19) n − n ≤ A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ + B Z M v d µ ; (II) Log-Sobolev inequality: for all v ∈ W , ( M ) such that k v k = 1 and all ǫ > , Z M v ln v d µ ≤ ǫ Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ − n ǫ + BA − ǫ + n nA e . Proof. I ⇒ II : The proof is a standard application of the Jensen’s inequality. Thederivation is almost same with (2.5). We only need to take y = ǫ nA instead, the log-Sobolev inequality will be obtained as desired. II ⇒ I : Notice that the LHS of log-Sobolev inequality is bounded from below for all v ∈ W , ( M ). Hence, the log-Sobolev inequality implies directly that for all v ∈ W , ( M ) A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ + B > . Since the log-Sobolev inequality holds for all ǫ >
0, the RHS of log-Sobolev inequalitycan be seen as a function of ǫ and reaches its minimum. Thus we have Z M v ln v d µ ≤ n (cid:20) A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ + B (cid:21) , (2.6) or all v ∈ W , ( M ) such that k v k = 1. Now we consider any function f ∈ W , ( M ). Bythe Kato’s inequality |∇| f || ≤ |∇ f | , we only need to prove the Sobolev inequality for allnonnegative functions. So we assume that f ≥
0. For the sake of conveniences, we denote W ( f ) = (cid:20) A Z M (cid:18) |∇ f | + 14 Sf (cid:19) d µ + B Z M f d µ (cid:21) . Taking v = f k f k in (2.6) yields Z M f ln (cid:18) f k f k (cid:19) d µ ≤ n k f k ln (cid:18) W ( f ) k f k (cid:19) . It is just ( LS q ) in Page 1067 of [3], where q = nn − . Using their method to treat the aboveestimate, we arrive at k f k ≤ W ( f ) θ k f k − θs , (2.7)where 0 < s < = θq + − θs . Next we need to define a family of functions f k for k ∈ Z by f k = min { ( f − k ) + , k } , where ( f − k ) + = max { f − k , } . From the definition of f k it is obvious to have thefollowing estimate for any p > pk µ { f ≥ k +1 } ≤ Z M f pk d µ ≤ pk µ { f ≥ k } . (2.8)Set a k = 2 qk µ { f ≥ k } . Combining (2.7) with (2.8), we derive a k +1 ≤ q ( k +1) − k Z M f k d µ ≤ q ( k +1) − k W ( f k ) θ k f k k − θ ) s ≤ q W ( f k ) θ a − θ ) s k . Consequently, by the H¨older inequality we have X k ∈ Z a k = X k ∈ Z a k +1 ≤ X k ∈ Z q W ( f k ) θ a − θ ) s k ≤ q X k ∈ Z W ( f k ) ! θ X k ∈ Z a s k ! − θ ≤ q X k ∈ Z W ( f k ) ! θ X k ∈ Z a k ! − θ ) s , where the last inequality follows from 0 < s <
2. This leads to X k ∈ Z a k ≤ q ( q − s )2 − s X k ∈ Z W ( f k ) ! q . (2.9) oreover, it follows from the definition of a k that Z M f q d µ = X k ∈ Z Z k ≤ f ≤ k +1 f q d µ ≤ X k ∈ Z q ( k +1) (cid:18) µ ( f ≥ k ) − µ ( f ≥ k +1 ) (cid:19) =(2 q − X k ∈ Z a k . (2.10)Now the rest of proof is only need to control the term W ( f k ) in (2.9). We have thefollowing key estimate. Claim . If A S + B ≥
0, then for any 0 ≤ f ∈ W , ( M ) we have X k ∈ Z W ( f k ) ≤ W ( f ) . Firstly, we note that Z M | f k | d µ =2 Z k +1 − k t p − µ ( f − k ≥ t )d t =2 Z k +1 k ( s − k ) p − µ ( f ≥ s )d s. Thus we have X k ∈ Z Z M | f k | d µ =2 X k ∈ Z Z k +1 k ( s − k ) µ ( f ≥ s )d s = sup k ∈ Z ( sup s ∈ [2 k , k +1 ] s − k s ) X k ∈ Z Z k +1 k sµ ( f ≥ s )d s ! ≤ Z M f d µ. Because of the assumption that A S + B ≥
0, we can denote ( A S + B )d µ as a new measure.By the same method as above we can derive X k ∈ Z Z M (cid:18) A S + B (cid:19) f k d µ ≤ Z M (cid:18) A S + B (cid:19) f d µ. (2.11)In addition, it also holds that X k ∈ Z Z M |∇ f k | d µ = X k ∈ Z Z k ≤ f ≤ k +1 |∇ f | d µ = Z M |∇ f | d µ. (2.12) ombining (2.11) with (2.12) yields X k ∈ Z W ( f k ) = X k ∈ Z (cid:18) A Z M (cid:18) |∇ f k | + 14 Sf k (cid:19) d µ + B Z M f k d µ (cid:19) ≤ A X k ∈ Z Z M |∇ f k | d µ + X k ∈ Z Z M (cid:18) A S + B (cid:19) f k d µ ≤ A Z M |∇ f | d µ + 12 Z M (cid:18) A S + B (cid:19) f d µ ≤ W ( f ) . Hence the proof of the claim has been completed. Now we can use the claim to finish theproof of lemma. One should be careful because the assumption in the claim does not haveto be true. But it has no effect on the final proof. Since M is a closed manifold, thereexists a nonnegative constant S such that S + S ≥ . By combining (2.9) and (2.10),we deduce Z M f q d µ ≤ (2 q − q ( q − s )2 − s X k ∈ Z W ( f k ) ! q ≤ (2 q − q ( q − s )2 − s X k ∈ Z (cid:18) A Z M (cid:18) |∇ f k | + S + S f k (cid:19) d µ + B Z M f k d µ (cid:19)! q ≤ (2 q − q ( q − s )2 − s (cid:18) W ( f ) + A Z M S f d µ (cid:19) q , (2.13)where the last inequality holds due to the claim. Note that the LHS of (2.6) is boundedfrom below for all v ∈ W , ( M ), it implies that λ = inf (cid:26) W ( f ) | Z M f d µ = 1 , f ∈ W , ( M ) (cid:27) > , i.e. Z M f d µ ≤ λ − W ( f ) , (2.14)for any f ∈ W , ( M ). Finally, the Sobolev inequality follows from (2.13) and (2.14). Wecomplete the proof of the lemma. (cid:3) Therefore, Theorem 1.1 follows directly from Lemma 2.2 and Lemma 2.3.
Proof of Theorem 1.1.
Note that our log-Sobolev inequality in Lemma 2.2 just has onemore term 4 tB A − than Lemma 2.3. Applying the same arguments with the second partin the proof of Lemma 2.3 to our log-Sobolev inequality, we have Z M f q d µ ( g ( t )) ≤ (2 q − q ( q − s )2 − s e tB A n − (cid:18) A S λ ( t ) (cid:19) q W ( f ) q , or any f ∈ W , ( M, g ( t )). Here W ( f ) = (cid:20) A Z M (cid:18) |∇ f | + 14 Sf (cid:19) d µ ( g ( t )) + B Z M f d µ ( g ( t )) (cid:21) ,λ ( t ) = inf (cid:26) W ( f ) | Z M f d µ ( g ( t )) = 1 , f ∈ W , ( M, g ( t )) (cid:27) , and other symbols are all same as above.By the definitions of F entropy and W ( f ), we can see that λ ( t ) is linearly related tothe first eigenvalue of F entropy. On the other hand, the first eigenvalue of F entropy ismonotone non-decreasing in time (cf. Proposition 3.3 [19]). So λ ( t ) is also non-decreasingin time. It follows that our uniform Sobolev inequality holds, and we get A ( t ) = (2 q − q q − s )2 − s e tB nA (cid:18) A S λ (0) (cid:19) A , and B ( t ) = (2 q − q q − s )2 − s e tB nA (cid:18) A S λ (0) (cid:19) B , where S depends on g and φ , and λ (0) depends on A , B , g and φ . Thus the proofof Theorem 1.1 is completed now. (cid:3) From Theorem 1.1 we can obtain a uniform Sobolev inequality along the harmonic-Ricciflow under the assumption of positive first eigenvalue of F entropy. Recall that λ is thefirst eigenvalue of F entropy with respect to the initial metric g , i.e. λ = inf k v k =1 Z M (4 |∇ v | + Sv )d µ ( g ) . The F entropy corresponds to Perelman’s F entropy for the Ricci flow introduced in [20].Similarly as the Ricci flow, the harmonic-Ricci flow can be interpreted as the gradient flowof F entropy modulo a pull-back by a family of diffeomorphisms. The eigenvalue of F entropy is a very powerful tool for the research on Ricci flow and Riemannian manifolds.More results can be found in [8, 15].Since ( M, g ) is a closed Riemannian manifold of dimension n ≥
3, the Sobolev inequalityholds, i.e. there exist positive constants A and B depending only on the initial metric g such that, for any v ∈ W , ( M ), (cid:18)Z M v nn − d µ ( g ) (cid:19) n − n ≤ A Z M |∇ v | d µ ( g ) + B Z M v d µ ( g ) . (2.15)If λ >
0, combining with Sobolev inequality (2.15), we have (cid:18)Z M v nn − d µ ( g ) (cid:19) n − n ≤ ˜ A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ ( g )where ˜ A depends only on initial metric g , φ and λ . This means that the assumptionof Sobolev inequality in Theorem 1.1 at initial time holds with B = 0. Hence, Theorem1.1 gives us the following result. orollary 2.4. Let ( g ( x, t ) , φ ( x, t ) ) be a solution of the harmonic-Ricci flow (1.1) in M n × [0 , T ) with initial value ( g , φ ). Assume that the first eigenvalue λ of F entropywith respect to the initial metric g is positive. Then there exists a positive constant A ,depending only on n , g , φ and λ , such that for all v ∈ W , ( M, g ( t )) , t ∈ [0 , T ) , itholds that (cid:18)Z M v nn − d µ (cid:0) g ( t ) (cid:1)(cid:19) n − n ≤ A Z M (cid:18) |∇ v | + 14 Sv (cid:19) d µ (cid:0) g ( t ) (cid:1) . (2.16) Remark . The analogous uniform Sobolev inequalities were given in [25] for the Ricciflow and [18] for the extended Ricci flow.3.
The proof of Theorem 1.2 and Theorem 1.3
In this section, we first prove the linear parabolic estimate under the harmonic-Ricci flowwith the help of the above Sobolev inequality (2.16) and Moser’s iteration. As a result,we derive an upper bound for the heat kernel, which is similar to the one known for thefixed metric case.
Proof of Theorem 1.2.
For any constant p ≥
1, it follows from (1.3) that Z M f p ∂ t f d µ − Z M f p ∆ f d µ ≤ a Z M f p +1 d µ, where the volume measure d µ = d µ ( g ( t )) for simplicity, and the same symbol will also beused in the rest of the proof. Integrating by parts, we have1 p + 1 Z M ∂ t f p +1 d µ + 4 p ( p + 1) Z M (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) d µ ≤ a Z M f p +1 d µ. Since ∂ t d µ = − S d µ and 4 p ≥ p + 1) for all p ≥
1, multiplying both sides by p + 1, weget ∂ t Z M f p +1 d µ + Z M Sf p +1 d µ + 2 Z M (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) d µ ≤ a ( p + 1) Z M f p +1 d µ. Notice that the condition S ≥
0, then we have ∂ t Z M f p +1 d µ + 12 Z M (cid:18) Sf p +1 + 4 (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) (cid:19) d µ ≤ a ( p + 1) Z M f p +1 d µ. (3.1)Next for any 0 < τ < σ < T we define ψ ( t ) = ≤ t ≤ τ , ( t − τ ) / ( σ − τ ) τ ≤ t ≤ σ , σ ≤ t ≤ T .
Multiplying (3.1) by ψ , we obtain ∂ t (cid:18) ψ Z M f p +1 d µ (cid:19) + 12 ψ Z M (cid:18) Sf p +1 + 4 (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) (cid:19) d µ ≤ [ a ( p + 1) ψ + ψ ′ ] Z M f p +1 d µ. ntegrating this with respect to t yieldssup σ ≤ t ≤ T Z M f p +1 d µ + 12 Z Tσ Z M (cid:18) Sf p +1 + 4 (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) (cid:19) d µ d t ≤ (cid:20) a ( p + 1) + 1 σ − τ (cid:21) Z Tτ Z M f p +1 d µ d t. By the assumption that S ≥ λ of F entropywith respect to the initial metric g is positive. Applying H¨older inequality, the aboveestimate and the Sobolev inequality (2.16), we deduce Z Tσ Z M f ( p +1)(1+ n ) d µ d t ≤ Z Tσ (cid:18)Z M f p +1 d µ (cid:19) n (cid:18)Z M f ( p +1) nn − d µ (cid:19) n − n d t ≤ sup σ ≤ t ≤ T (cid:18)Z M f p +1 d µ (cid:19) n Z Tσ A Z M (cid:18) Sf p +1 + 4 (cid:12)(cid:12)(cid:12) ∇ f p +12 (cid:12)(cid:12)(cid:12) (cid:19) d µ d t ≤ n A (cid:20) a ( p + 1) + 1 σ − τ (cid:21) n (cid:18)Z Tτ Z M f p +1 d µ d t (cid:19) n . Set H ( p, τ ) = (cid:18)Z Tτ Z M f p d µ d t (cid:19) p , for any p ≥ < τ < T . So we get H ( p (1 + 2 n ) , σ ) ≤ (cid:16) n A (cid:17) p (1+ 2 n ) (cid:18) ap + 1 σ − τ (cid:19) p H ( p, τ ) . (3.2)Now we fix 0 < t < t < T , p ≥
2. Let χ = 1 + n , p k = p χ k , τ k = t + (1 − χ k )( t − t ).Then it follows from (3.2) that H ( p k +1 , τ k +1 ) ≤ (cid:16) n A (cid:17) pk +1 (cid:18) ap k + χ k t − t χχ − (cid:19) pk H ( p k , τ k ) . Hence by iteration, we arrive at H ( p m +1 , τ m +1 ) ≤ (cid:16) n A (cid:17) m P k =0 1 pk +1 (cid:18) ap + 1 t − t χχ − (cid:19) m P k =0 1 pk χ m P k =0 kpk H ( p , τ ) . Letting m → ∞ , we obtain H ( p ∞ , τ ∞ ) ≤ C (cid:20) ap + n + 22( t − t ) (cid:21) n +22 p H ( p , τ ) . for all p ≥
2. This meanssup ( x,t ) ∈ M × [ t ,T ] | f ( x, t ) | ≤ (cid:18) C a + C t − t (cid:19) n +22 p (cid:18)Z Tt Z M f p d µ d t (cid:19) p , (3.3) here C and C are both positive constants depending on dimension n , p , initial metric g , φ and λ . Moreover, for 0 < p <
2, we set h ( s ) = sup ( x,t ) ∈ M × [ s,T ] | f ( x, t ) | . Combining Young inequality with (3.3), we deduce h ( t ) ≤ (cid:18) C a + C t − t (cid:19) n +24 (cid:18)Z Tt Z M f d µ d t (cid:19) ≤ h ( t ) − p (cid:18) C a + C t − t (cid:19) n +24 (cid:18)Z Tt Z M f p d µ d t (cid:19) ≤ h ( t ) + (cid:18) C a + C t − t (cid:19) n +22 p (cid:18)Z Tt Z M f p d µ d t (cid:19) p Now we use the iteration method again. Fix 0 < t < t < T , for some 0 < θ <
1, we let x k = t − (1 − θ k )( t − t ). Then by iteration h ( t ) = h ( x ) ≤ k h ( x k ) + (cid:18)Z Tt Z M f p d µ d t (cid:19) p k − X i =0 i (cid:18) C a + C x i − x i +1 (cid:19) n +22 p ≤ k h ( x k ) + (cid:18)Z Tt Z M f p d µ d t (cid:19) p (cid:18) C a + C t − t (cid:19) n +22 p k − X i =0 (cid:16) θ n +22 p (cid:17) − i Choose 0 < θ < θ n +22 p >
1, that is, < θ n +22 p <
1. Taking k → ∞ , we have h ( t ) ≤ (cid:18) C a + C t − t (cid:19) n +22 p (cid:18)Z Tt Z M f p d µ d t (cid:19) p , (3.4)for all 0 < p < < t < t < T . By (3.3) and (3.4) together, as t →
0, it followsthat h ( t ) ≤ (cid:18) C a + C t (cid:19) n +22 p (cid:18)Z T Z M f p d µ d t (cid:19) p , ∀ p > C and C are both positive constants depending on dimension n , p , initial metric g , φ and λ . Thus we complete the proof now. (cid:3) Note that the heat equation (1.2) is a linear parabolic equation. The above proof forTheorem 1.2 can be applied to the heat equation almost verbatim. Thus it is not difficultto get the following corollary, which will be used to determine the upper bound of theheat kernel later.
Corollary 3.1.
Assume that ( g ( x, t ) , φ ( x, t ) ) is a smooth solution to the harmonic-Ricciflow (1.1) in M n × [0 , T ] with the initial value ( g , φ ) and S ≥ at the initial time. Let u be a nonnegative smooth solution to the heat equation (1.2) on M × [0 , T ] . Then forany ≤ s < t ≤ T and p > we have sup x ∈ M | u ( x, t ) | ≤ C ( t − s ) n +22 p (cid:18)Z ts Z M u ( x, τ ) p d µ d τ (cid:19) p , here C is a positive constant depending on dimension n , p , g , φ and λ . In fact, nonnegativity of S in the conditions of Theorem 1.2 can be removed. At this timethe parabolic estimate will also depend on the negative lower bound of S . By the similararguments of Theorem 1.2 we can obtain the following estimate. Corollary 3.2.
Assume that ( g ( x, t ) , φ ( x, t ) ) is a smooth solution to the harmonic-Ricciflow (1.1) in M n × [0 , T ] with initial value ( g , φ ), S ≥ − S for a nonnegative constant S at the initial time, and the first eigenvalue λ of F entropy with respect to g is positive.Let f be a nonnegative Lipschitz continuous function on M × [0 , T ] satisfying ∂ t f ≤ ∆ f + af on M × [0 , T ] in the weak sense, where a ≥ . Then we have for any < t ≤ T and p > x ∈ M | f ( x, t ) | ≤ (cid:18) C S + C a + C t (cid:19) n +22 p (cid:18)Z T Z M f p d µ d t (cid:19) p , where C , C and C are all positive constants depending on dimension n , p , g , φ and λ . Now we turn to control the heat kernel, which is easy to carry out by means of using theresult in Corollary 3.1.
Proof of Theorem 1.3.
By the definition of heat kernel, we know the fact ∂ t G ( x, t ; y, s ) − ∆ x G ( x, t ; y, s ) = 0 . Combining with the assumption of S ≥
0, we have ∂ t Z M G ( x, t ; y, s )d µ ( x, t ) = Z M (cid:2) ∆ x G ( x, t ; y, s ) − SG ( x, t ; y, s ) (cid:3) d µ ( x, t ) ≤ . It implies that the above integral of heat kernel is non-increasing in t . So we derive Z M G ( x, t ; y, s )d µ ( x, t ) ≤ Z M G ( x, s ; y, s )d µ ( x, s ) = 1 , for ∀ ≤ s < t ≤ T . Therefore, by Corollary 3.1, it follows that G ( x, t ; y, s ) ≤ C ( t − s ) n +22 Z ts Z M G ( x, τ ; y, s )d µ ( x, τ )d τ ≤ C ( t − s ) n , where C is a positive constant depending on dimension n , g , φ and λ . (cid:3) Thanks to Corollary 3.2, the same upper bound of heat kernel can be given without theassumption of the nonnegativity of S but at the cost of the dependence on the upperbound for time. Corollary 3.3.
Assume that ( g ( x, t ) , φ ( x, t ) ) is a smooth solution to the harmonic-Ricciflow (1.1) in M n × [0 , T ] with initial value ( g , φ ) and the first eigenvalue λ of F entropy ith respect to g is positive. Let G ( x, t ; y, s ) be the heat kernel. Then for ∀ x, y ∈ M and ∀ ≤ s < t ≤ T , we have G ( x, t ; y, s ) ≤ C ( t − s ) n , where C is a positive constant, which depends on dimension n , g , φ , λ and T .Proof. By Theorem 1.3, we only need to prove the estimate for the case that S has negativeminimum at the initial time. We can assume that S = − min x ∈ ( M,g ) S ( x, >
0. From thedefinition of heat kernel and Corollary 3.2, we havesup x ∈ M | G ( x, t ; y, s ) | ≤ (cid:18) C S + C t − s (cid:19) n +22 Z ts Z M G ( x, τ ; y, s )d µ ( x, τ )d τ, (3.5)for ∀ ≤ s < t ≤ T . In addition, since S > − S for any time t ∈ [0 , T ], we deduce ∂ t Z M G ( x, t ; y, s )d µ ( x, t ) = Z M (cid:2) ∆ x G ( x, t ; y, s ) − SG ( x, t ; y, s ) (cid:3) d µ ( x, t ) ≤ S Z M G ( x, t ; y, s )d µ ( x, t ) . Integrating from s to τ gives Z M G ( x, τ ; y, s )d µ ( x, τ ) ≤ e S ( τ − s ) Z M G ( x, s ; y, s )d µ ( x, s ) = e S ( τ − s ) , for ∀ ≤ s < τ ≤ T . Therefore, combining with (3.5), we conclude that G ( x, t ; y, s ) ≤ (cid:18) C S + C t − s (cid:19) n +22 Z ts e S ( τ − s ) d τ ≤ C ( t − s ) n , where the last inequality follows from 0 < t − s ≤ T and C is a positive constant dependingon dimension n , g , φ , λ and T . (cid:3) Acknowledgements.
This work was carried out while the authors were visiting North-western Univerisity. We would like to thank Professor Valentino Tosatti and ProfessorBen Weinkove for hospitality and helpful discussions. We also thank Wenshuai Jiang forsome useful conversations.
References [1]
M. Bailesteanu , Bounds on the heat kernel under the Ricci flow,
Proc. Amer. Math. Soc. (2012), 691–700.[2]
M. Bailesteanu and H. Tran , Heat kernel estimates under the Ricci-harmonic map flow, arXiv:math.DG/1310.1619 .[3]
D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste , Sobolev inequalities in disguise,
Indiana Univ. Math. J. (1995), 1033–1047.[4] H. D. Cao and X. P. Zhu , A complete proof of the Poincar´e and Geometrization conjectures—Application of the Hamilton-Perelman theory of the Ricci flow,
Asian J. Math. (2006), 165–492.[5] B. Chow, P. Lu and L. Ni , Hamiltons Ricci flow,
Graduate Studies in Mathematics , vol. 77,American Mathematical Society, Providence, RI, 2006. S. W. Fang , Differential Harnack inequalities for heat equations with potentials under the BernhardList’s flow,
Geom. Dedicata (2012), 11–22.[7]
S. W. Fang , Differential Harnack inequalities for backward heat equations with potentials underan extended Ricci flow,
Adv. Geom. (2013), 741–755.[8] S. W. Fang, H. F. Xu and P. Zhu , Evolution and monotonicity of eigenvalues under the Ricciflow,
Sci. China Math. (2015), doi: 10.1007/s11425-014-4943-7.[9] S. W. Fang and P. Zhu , Differential Harnack estimates for backward heat equations with potentialsunder geometric flows,
Commun. Pur. Appl. Anal. , to appear.[10]
R. S. Hamilton , Three-mainfolds with positive Ricci curvature,
J. Differ. Geom. (1982), 255–306.[11] R. S. Hamilton , Four manifolds with positive Ricci curvature,
J. Differ. Geom. (1986), 153–179.[12] S. Y. Hsu , Uniform Sobolev inequalities for manifolds evolving by Ricci flow, arXiv:math.DG/0708.0893 .[13]
W. S. Jiang , Bergman kernel along the K¨ahler-Ricci flow and Tian’s conjecture,
J. reine angew.Math. (2014), doi: 10.1515/crelle-2014-0015.[14]
S. L. Kuang and Q. S. Zhang , A gradient estimate for all positive solutions of the conjugate heatequation under Ricci flow,
J. Funct. Anal. (2008), 1008–1023.[15]
J. F. Li , Eigenvalues and energy functionals with monotonicity formulae under Ricci flow,
Math.Ann. (2007), 927–946.[16]
P. Li and S. T. Yau , On the parabolic kernel of the Schr¨odinger operator,
Acta Math. (1986),153–201.[17]
B. List , Evolution of an extended Ricci flow system,
Comm. Anal. Geom. (2008), 1007–1048.[18] X. G. Liu and K. Wang , A Gaussian upper bound of the conjugate heat equation along anextended Ricci flow, arXiv:math.DG/1412.3200 .[19]
R. M¨uller , Ricci flow coupled with harmonic map flow,
Ann. Sci. Ecole Norm. S. (2012),101–142.[20] G. Perelman , The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/ 0211159 .[21]
J. Wang , Global heat kernel estimates,
Pacific J. Math. (1997), 377–398.[22]
R. G. Ye , Curvature estimates for the Ricci flow I,
Calc. Var. (2008), 417–437.[23] R. G. Ye , The logarithmic Sobolev inequality along the Ricci flow, arXiv:math.DG/0707.2424 .[24]
Q. S. Zhang , Some gradient estimates for the heat equation on domain for an equation by Perelman,
International Mathematics Research Notices (2006), article id: 92314, 1–39.[25]
Q. S. Zhang , A uniform Sobolev inequality under Ricci flow,
International Mathematics ResearchNotices (2007), article id: rnm056, 1–17.[26]
Q. S. Zhang , Sobolev inequalities, heat kernels under Ricci flow and Poincar´e conjecture, CRCPress, Boca Raton, FL, 2010.[27]
A. Q. Zhu ,Differential Harnack inequalities for the backward heat equation with potential underthe harmonic-Ricci flow,
J. Math. Anal. Appl. (2013), 502–510.