Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds
aa r X i v : . [ m a t h . DG ] O c t Hamilton differential Harnack inequality and W -entropy for WittenLaplacian on Riemannian manifolds Songzi Li ∗ , Xiang-Dong Li † April 24, 2018
Abstract . In this paper, we prove the Hamilton differential Harnack inequalityfor positive solutions to the heat equation of the Witten Laplacian on completeRiemannian manifolds with the CD ( − K, m )-condition, where m ∈ [ n, ∞ ) and K ≥ W -entropy and provethe W -entropy formula for the fundamental solution of the Witten Laplacianon complete Riemannian manifolds with the CD ( − K, m )-condition and oncompact manifolds equipped with ( − K, m )-super Ricci flows.
MSC2010 Classification : primary 53C44, 58J35, 58J65; secondary 60J60, 60H30.
Keywords : Hamilton differential Harnack inequality, W -entropy, super Ricci flows. Differential Harnack inequality is an important tool in the study of geometric PDEs. Let M be an n dimensional complete Riemannian manifold, u a positive solution to the heatequation ∂ t u = ∆ u. (1)In their famous paper [8], Li and Yau proved that if Ric ≥ − K , where K ≥ α > |∇ u | u − α ∂ t uu ≤ nα t + nα K √ α − . (2)In particular, if Ric ≥
0, then taking α →
1, the Li-Yau differential Harnack inequalityholds |∇ u | u − ∂ t uu ≤ n t . (3)In [6], Hamilton proved a dimension free differential Harnack inequality on compactRiemannian manifolds with Ricci curvature bounded from below. More precisely, if M is acompact Riemannian manifold with Ric ≥ − K, then, for any positive and bounded solution u to the heat equation (1), it holds |∇ u | u ≤ (cid:18) t + 2 K (cid:19) log( A/u ) , ∀ x ∈ M, t > , (4) ∗ Research partially supported by the China Scholarship Council and a Postdoctoral Fellowship of BeijingNormal University. † Research supported by NSFC No. 11371351, Key Laboratory RCSDS, CAS, No. 2008DP173182, and aHundred Talents Project of AMSS, CAS. A := sup { u ( t, x ) : x ∈ M, t ≥ } . Indeed, the same result holds on complete Rie-mannian manifolds with Ricci curvature bounded from below. Under the same condition Ric ≥ − K , Hamilton also proved the following differential Harnack inequality for any posi-tive solution to the heat equation (1) |∇ u | u − e Kt ∂ t uu ≤ n t e Kt . (5)In particular, when K = 0, the above inequality reduces to the Li-Yau Harnack inequality (3)on complete Riemannian manifolds with non-negative Ricci curvature. Moreover, Hamilton[6] proved that, on compact Riemannian manifolds with Ric ≥ − K , then any positive andbounded solution of the heat equation ∂ t u = ∆ u with 0 < u ≤ A satisfies ∂ t uu + |∇ u | u ≤ K − e − Kt [ n + 4 log( A/u )] , ∀ t ≥ . (6)On the other hand, Perelman [22] reformulated the Ricci flow as the gradient flow ofthe F -functional, where F ( g, f ) = R M ( R + |∇ f | ) e − f dv is defined on the product space ofRiemannian metrics and C ∞ -functions equipped with the standard L -Riemannian metricwith the constraint that e − f dv does not change, where R is the scalar curvature of g . Hethen introduced the W -entropy functional and proved its monotonicity along the conjugateequation coupled with the Ricci flow. The F -functional has been used by Perelman to char-acterize the steady gradient Ricci solitons, and the W -entropy has been used to characterizethe shrinking gradient Ricci solitons. As an application of the W -entropy formula, Perelman[22] proved the non local collapsing theorem for the Ricci flow, which plays an importantrˆole for ruling out cigars, one part of the singularity classification for the final resolution ofthe Poincar´e conjecture and geometrization conjecture.Since Perelman’s preprint [22] was posted on Arxiv in 2002, many people have studied the W -like entropy for other geometric flows on Riemannian manifolds. In [20, 21], Ni proved the W -entropy formula for the heat equation ∂ t u = ∆ u on compact and complete Riemannianmanifolds with non-negative Ricci curvature, where ∆ denotes the usual Laplace-Beltramioperator on Riemannian manifolds. In [18], Li and Xu extended Ni’s W -entropy formulato the heat equation ∂ t u = ∆ u on complete Riemannian manifolds with Ricci curvaturebounded from below by a negative constant.From [22, 20, 18, 10, 12, 13], it has been known that there is a close connection betweenthe differential Harnack inequality and the W -entropy for the heat equation on Riemannianmanifolds. To see this link, let ( M, g ) be a complete Riemannian manifold with boundedgeometry condition, u be a positive solution to the heat equation ∂ t u = ∆ u . As in [20, 21],let H n ( u ( t )) = − Z M u log udv − n πt ) + 1) . (7)Then ddt H n ( u ( t )) = Z M (cid:20) |∇ u | u − ∂ t uu − n t (cid:21) udv. Suppose that (
M, g ) is a complete Riemannian manifold with non-negative Ricci curvature.Then the Li-Yau Harnack inequality (3) holds. This yields ddt H n ( u ( t )) ≤ . Let W n ( u ( t )) = ddt ( tH n ( u ( t )) .
2n [20, 21], Ni proved that ddt W n ( u ( t )) = − t Z M (cid:20)(cid:12)(cid:12)(cid:12) ∇ log u + g t (cid:12)(cid:12)(cid:12) + Ric ( ∇ log u, ∇ log u ) (cid:21) udv. We now introduce some notations and definitions to develop the main part of this paper.Let (
M, g ) be a complete Riemannian manifold, φ ∈ C ( M ) and dµ = e − φ dv , where v is theRiemannian volume measure on ( M, g ). The Witten Laplacian acting on smooth functionsis defined by L = e φ div( e − φ ∇ ) = ∆ − ∇ φ · ∇ . For any u, v ∈ C ∞ ( M ), the integration by parts formula holds Z M h∇ u, ∇ v i dµ = − Z M Luvdµ = − Z M uLvdµ. Thus, L is the infinitesimal generator of the Dirichlet form E ( u, v ) = Z M h∇ u, ∇ v i dµ, u, v ∈ C ∞ ( M ) . In [1], Bakry and Emery proved that for all u ∈ C ∞ ( M ), L |∇ u | − h∇ u, ∇ Lu i = 2 |∇ u | + 2 Ric ( L )( ∇ u, ∇ u ) , (8)where Ric ( L ) = Ric + ∇ φ is now called the infinite dimensional Bakry-Emery Ricci curvature associated with theWitten Laplacian L . For m ∈ [ n, ∞ ), the m -dimensional Bakry-Emery Ricci curvatureassociated with the Witten Laplacian L is defined by Ric m,n ( L ) = Ric + ∇ φ − ∇ φ ⊗ ∇ φm − n . In view of this, we have (see [9]) L |∇ u | − h∇ u, ∇ Lu i ≥ | Lu | m + 2 Ric m,n ( L )( ∇ u, ∇ u ) . Here we only define
Ric m,n ( L ) for m = n when φ is a constant. By definition, we have Ric ( L ) = Ric ∞ ,n ( L ) . Following [1], we say that (
M, g, µ ) satisfies the curvature-dimension CD ( K, m )-condition for a constant K ∈ R and m ∈ [ n, ∞ ] if and only if Ric m,n ( L ) ≥ Kg.
Inspired by Perelman’s introduction of the modified Ricci flow ∂ t g = − Ric + ∇ φ ) in[22], we define the ( K, m )-(Perelman) Ricci flow and (
K, m )-super (Perelman) Ricci flowsas follows. We call a manifold (
M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) equipped with a family of timedependent Riemann metrics g ( t ) and C -potentials φ ( t ) a ( K, m )-(Perelman) Ricci flow if12 ∂g∂t + Ric m,n ( L ) = Kg, ∀ t ∈ (0 , T ] , Here the word “ CD ” means “curvature-dimension”. M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) a ( K, m )-super (Perelman) Ricci flow if12 ∂g∂t + Ric m,n ( L ) ≥ Kg, ∀ t ∈ (0 , T ] . See also our previous paper [13] and [14, 15, 16]. When φ is a constant and m = n , the( K, n )-(Perelman) Ricci flow is indeed the Hamilton K -Ricci flow12 ∂g∂t + Ric = Kg, ∀ t ∈ (0 , T ] , and a ( K, n )-super (Perelman) Ricci flow is a Hamilton K -super Ricci flow12 ∂g∂t + Ric ≥ Kg, ∀ t ∈ (0 , T ] . While when m = ∞ , the ( K, ∞ )-(Perelman) Ricci flow is indeed the following extension ofthe modified Ricci flow introduced by Perelman [22] (where K = 0)12 ∂g∂t + Ric ( L ) = Kg, ∀ t ∈ (0 , T ] , and a ( K, ∞ )-super (Perelman) Ricci flow reads as follows12 ∂g∂t + Ric ( L ) ≥ Kg, ∀ t ∈ (0 , T ] . We would like to point out that the notion of super Ricci flows has been also independentlyintroduced by K.-T. Sturm on time-dependent metric measure spaces [24]. See also Kopfer-Sturm [7].In [9], the Li-Yau Harnack inequality (2) has been extended to positive solutions of theheat equation of the Witten Laplacian ∂ t u = Lu (9)on complete Riemannian manifolds with the CD ( − K, m )-condition, i.e., the m -dimensionalBakry-Emery Ricci curvature associated with L satisfies Ric m,n ( L ) ≥ − Kg , where m ∈ [ n, ∞ ) and K ≥
0. In particular, on complete Riemannian manifolds with the CD (0 , m )-condition, the classical Li-Yau Harnack inequality (3) has been extended to positive solutionsto the heat equation (9) (see [10]) |∇ u | u − ∂ t uu ≤ m t . (10)In [12], an improved version of the Hamilton Harnack inequality (4) has been established forpositive and bounded solutions to the heat equation (9) on complete Riemannian manifoldswith the CD ( − K, ∞ )-condition, where K ≥ W -entropy formula has been also extended to theheat equation of the Witten Laplacian (9) on complete Riemannian manifolds with non-negative m -dimensional Bakry-Emery Ricci curvature condition. More precisely, let ( M, g )be a complete Riemannian manifold with bounded geometry condition, u a positive solutionto the heat equation (9) of the Witten Laplacian on ( M, g, µ ). Let H m ( u ( t )) = − Z M u log udµ − m πt ) + 1) . (11)Then ddt H m ( u ( t )) = Z M (cid:20) |∇ u | u − ∂ t uu − m t (cid:21) udµ. M, g ) is a complete Riemannian manifold with the CD (0 , m )-condition. Thenthe Li-Yau Harnack inequality (10) yields ddt H m ( u ( t )) ≤ . Let W m ( u ( t )) = ddt ( tH m ( u ( t ))) . (12)By [10, 12], we have the W -entropy formula for the heat equation of the Witten Laplacian ddt W m ( u ( t )) = − t Z M (cid:20)(cid:12)(cid:12)(cid:12) ∇ log u + g t (cid:12)(cid:12)(cid:12) + Ric m,n ( L )( ∇ log u, ∇ log u ) (cid:21) udµ − tm − n Z M (cid:18) ∇ log u · ∇ φ − m − n t (cid:19) udµ. (13)In particular, ddt W m ( u ( t )) ≤ Ric m,n ( L ) ≥
0. Moreover, the above definition formu-las (11) and (12) indicate also the close connection between the extended Li-Yau Harnackinequality (10) and the W -entropy on complete Riemannian manifolds with the CD (0 , m )-condition. Moreover, a rigidity theorem for W m was also proved in [10] on complete Rieman-nian manifolds with the CD (0 , m )-condition. See also [11] for the W -entropy formula for theFokker-Planck equation on complete Riemannian manifolds with the CD (0 , m )-condition.In our previous papers [13, 16], we proved the W -entropy formula for the heat equationof the Witten Laplacian on complete Riemannian manifolds with the CD ( − K, m )-condition, m ∈ [ n, ∞ ) and K ≥
0. These extend Ni and Li-Xu’s results from the standard case of heatequation of the Laplace-Beltrami operator on complete Riemannian manifolds with Riccicurvature condition to the general case of the heat equation of the Witten Laplacian oncomplete weighted Riemannian manifolds with suitable m -dimensional Bakry-Emery Riccicurvature condition. In [13], when m ∈ N , we gave a direct proof of the W -entropy formulafor the Witten Laplacian by applying Ni’s or Li-Xu’s W -entropy formula for the usualLaplacian to M × S m − n equipped with a suitable warped product Riemannian metric, andgave a natural geometric interpretation of the W -entropy formula for the heat equation of theWitten Laplacian. In [13], we have also proved the W -entropy formula for the heat equationof time dependent Witten Laplacian on Riemannian manifolds equipped with ( K, m )-superRicci flows, where m ∈ [ n, ∞ ] and K ∈ R . More precisely, for K = 0, let ( M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) be a compact manifolds equipped with a family of time dependent metrics g ( t ) and C -potentials φ ( t ), t ∈ [0 , T ] such that dµ = e − φ dv is independent of t (which is equivalentto the conjugate heat equation ∂ t φ = Tr( ∂ t g )), then the W -entropy defined by (11) and(12) for positive solution to the heat equation (9) of the time dependent Witten Laplacian L = ∆ g ( t ) − ∇ g ( t ) φ ( t ) · ∇ g ( t ) satisfies ddt W m ( u ( t )) = − t Z M (cid:20)(cid:12)(cid:12)(cid:12) ∇ log u + g t (cid:12)(cid:12)(cid:12) + (cid:18) ∂g∂t + Ric m,n ( L ) (cid:19) ( ∇ log u, ∇ log u ) (cid:21) udµ − tm − n Z M (cid:18) ∇ log u · ∇ φ − m − n t (cid:19) udµ. (14)In particular, if ( M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) is a (0 , m )-super Ricci flow in the sense that12 ∂g∂t + Ric m,n ( L ) ≥ , ∂φ∂t = 12 Tr (cid:18) ∂g∂t (cid:19) , then W m ( u ( t )) is decreasing in time t on [0 , T ]. For general case K ≥ m ∈ [ n, ∞ ], see[13, 16]. 5he purpose of this paper is threefolds. First, we extend the Hamilton differential Har-nack inequalities (5) and (6) to positive solutions of the heat equation (9) of the WittenLaplacian on complete weighted Riemannian manifolds with the CD ( − K, m ) condition.Second, we use Hamilton’s Harnack inequality to introduce a new W -entropy and prove anew W -entropy formula to positive solutions of the heat equation (9) of the Witten Lapla-cian on complete weighted Riemannian manifolds with the CD ( − K, m ) condition. We alsocompare our new W -entropy with the W -entropy we introduced in [13] on complete Rieman-nian manifolds with the CD ( − K, m )-condition. Finally, we extend the W -entropy formulato the heat equation (9) associated with the time dependent Witten-Laplacian on compactmanifolds equipped with ( K, m )-super Ricci flows. As mentioned above, by previous worksin [20, 10, 12, 13], there exists an essential link between the W -entropy and the Li-Yau Har-nack inequality (10) for the heat equation of the Witten Laplacian on complete Riemannianmanifolds satisfying the CD (0 , m )-condition. Our result indicates that, when m ∈ [ n, ∞ )and K ≥
0, there still exists an essentially deep connection between the W -entropy and theHamilton differential Harnack inequality (5) for the heat equation of the Witten Laplacianon complete Riemannian manifolds with the CD ( − K, m )-condition.
Now we are in a position to state our main results. Our first result extends the Hamiltondifferential Harnack inequality (5) to the heat equation of the Witten Laplacian on completeRiemannian manifolds with the CD ( − K, m )-condition.
Theorem 2.1
Let ( M, g ) be a complete Riemannian manifold and φ ∈ C ( M ) . Supposethat there exist some constants m ∈ [ n, ∞ ) and K ≥ such that Ric m,n ( L ) ≥ − K . Let u be a positive solution to the heat equation (9) . Then the Hamilton differential Harnackinequality holds |∇ u | u − e Kt ∂ t uu ≤ m t e Kt . (15) In particular, if
Ric m,n ( L ) ≥ , then the Li-Yau differential Harnack inequality holds |∇ u | u − ∂ t uu ≤ m t . Integrating the differential Harnack inequality along the geodesic on the space time,Theorem 2.1 implies the following Harnack inequality.
Theorem 2.2
Under the same condition and notation as in Theorem 2.1, for all x, y ∈ M , < τ < T , we have u ( x, τ ) u ( y, T ) ≤ (cid:18) Tτ (cid:19) m/ exp (cid:26) e Kτ [1 + 2 K ( T − τ )] d ( x, y ) T − τ + m e KT − e Kτ ] (cid:27) . The following result extends Hamilton’s estimate (6) to the heat equation of the WittenLaplacian on complete Riemannian manifolds with the CD ( − K, m )-condition.
Theorem 2.3
Let ( M, g ) be a complete Riemannian manifold with bounded Riemanniancurvature tensor, φ ∈ C ( M ) such that ∇ φ and ∇ φ are uniformly bounded on M . Supposethat there exist some constants m ∈ [ n, ∞ ) and K ≥ such that Ric m,n ( L ) ≥ − K . Then forany bounded and positive solution u to the heat equation (9) with A = sup { u ( x, t ) , ( x, t ) ∈ M × [0 , T ] } < ∞ , it holds ∂ t uu + |∇ u | u ≤ K − e − Kt [ m + 4 log( A/u )] , ∀ t ∈ [0 , T ] . (16)6 n particular, for t ∈ [0 , T ] , we have ∂ t uu ≤ (cid:18) K + 1 t (cid:19) [ m + 4 log( A/u )] . (17)We would like to mention that, as was pointed out in the report of an anonymous referee,the above estimates are central tools in the study of the classical heat equation as well asRicci flow and Ricci solitons, so in principle there are similar applications waiting to followin this area for the heat equation of the Witten Laplacian as well as ( K, m )-Ricci flow and(
K, m )-Ricci solitons.As an application of Theorem 2.1 and Theorem 2.3, we can derive the following bound forthe time derivative of the logarithm of the heat kernel of the Witten Laplacian on completeRiemannian manifolds with the CD ( − K, m )-condition, we have − m t e Kt ≤ ∂ t log p t ( x, y ) ≤ (cid:18) K + 1 t (cid:19) m + 4 log sup x ∈ M p t ( x, y )inf x ∈ M p t ( x, y ) . Using the upper bound and lower bound estimates of the heat kernel p t ( x, y ) on completeRiemannian manifolds with the CD ( − K, m )-condition obtained in [9, 10, 12], we can derivethe following estimate, which seems new in the literature.
Theorem 2.4
Under the same condition and notation as in Theorem 2.3, for all fixed
T > and t ∈ (0 , T ] , we have − m t e Kt ≤ ∂ t log p t ( x, y ) ≤ C m,n,K,T (cid:18) √ t + d ( x, y ) t (cid:19) , where p t ( x, y ) denotes the heat kernel of the Witten Laplacian L with respect to µ on ( M, g ) . Remark 2.5
In [12], it has been proved that under the condition (
M, g ) is a completeRiemannian manifold with bounded geometry condition (i.e., the Riemannian curvaturetensor as well its k -th covariant derivatives are uniformly bounded up to the 3-rd order), φ ∈ C ( M ) such that ∇ k φ are uniformly bounded on M for 1 ≤ k ≤
4, then | ∂ t log p t ( x, y ) | ≤ C m,n,K,T (cid:18) √ t + d ( x, y ) t (cid:19) . While Theorem 2.4 need only to assume the Riemannian curvature tensor Riem is uniformlybounded,
Ric m,n ( L ) ≥ − Kg and φ ∈ C ( M ) such that ∇ φ and ∇ φ are uniformly boundedon M .The following result indicates the close connection between the Hamiton differentialHarnack inequality (5) and the W -entropy for the heat equation of the Witten Laplacian oncomplete Riemannian manifolds with the CD ( − K, m )-condition. When K = 0, it reducesto the W -entropy formula (13) for the heat equation of the Witten Laplacian on completeRiemannian manifolds with the CD (0 , m )-condition. Theorem 2.6
Let ( M, g ) be a complete Riemannian manifold with the bounded geometrycondition and φ ∈ C ( M ) such that ∇ k φ are uniformly bounded on M for ≤ k ≤ . Let u be the heat kernel of the Witten Laplacian L = ∆ − ∇ φ · ∇ . Let H m,K ( u, t ) = − Z M u log udµ − Φ m,K ( t ) , here Φ m,K ∈ C ((0 , ∞ ) , R ) satisfies Φ ′ m,K ( t ) = m t e Kt , ∀ t > . Define the W -entropy by the Boltzmann formula W m,K ( u, t ) = ddt ( tH m,K ( u, t )) . Then ddt W m,K ( u, t ) = − t Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) K t (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) udµ − t Z M ( Ric m,n ( L ) + Kg ) ( ∇ log u, ∇ log u ) udµ − tm − n Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ φ · ∇ log u − ( m − n )(1 + Kt )2 t (cid:12)(cid:12)(cid:12)(cid:12) udµ − m t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . In particular, if
Ric m,n ( L ) ≥ − Kg , then, for all t > , we have ddt W m,K ( u, t ) ≤ − m t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . Moreover, the equality holds at some time t = t > if and only if ( M, g, φ ) is a fixed point ofthe ( − K, m ) -Ricci flow, called ( − K, m ) -Ricci soliton or ( − K, m ) -quasi-Einstein manifold, Ric m,n ( L ) = − Kg, the potential function f = − log u satisfies the shrinking soliton equation with respect to Ric m,n ( L ) , i.e., Ric m,n ( L ) + 2 ∇ f = gt , and moreover ∇ φ · ∇ f = − ( m − n )(1 + Kt )2 t . The following result extends Theorem 2.6 to the heat equation of the time dependentWitten Laplacian on compact manifolds equipped with ( − K, m )-super Ricci flows. When K = 0, it is the W -entropy formula (14) on the (0 , m )-super Ricci flows, which was provedin our previous paper [13]. Theorem 2.7
Let ( M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) be a compact manifold with a family of Rieman-nian metrics and C ∞ -potentials ( g ( t ) , φ ( t ) , t ∈ [0 , T ]) . Suppose that ∂φ∂t = 12 Tr (cid:18) ∂g∂t (cid:19) . Let u be a positive solution to the heat equation (9) of the time dependent Witten Laplacian L = ∆ g ( t ) − ∇ g ( t ) φ ( t ) · ∇ g ( t ) . et H m,K ( u, t ) and W m,K ( u, t ) be as in Theorem 2.6. Then ddt W m,K ( u, t ) = − t Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) K t (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) udµ − t Z M (cid:18) ∂g∂t + Ric m,n ( L ) + Kg (cid:19) ( ∇ log u, ∇ log u ) udµ − tm − n Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ φ · ∇ log u − ( m − n )(1 + Kt )2 t (cid:12)(cid:12)(cid:12)(cid:12) udµ − m t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . In particular, if ( M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) is a compact manifolds equipped with a ( − K, m ) -super Ricci flow in the sense that ∂g∂t + Ric m,n ( L ) ≥ − K, ∂φ∂t = 12 Tr (cid:18) ∂g∂t (cid:19) , then for all t ∈ (0 , T ] , we have ddt W m,K ( u, t ) ≤ − m t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . Moreover, the equality holds on (0 , T ] if and only if ( M, g ( t ) , φ ( t ) , t ∈ [0 , T ]) is a ( − K, m ) -Ricci flow in the sense that ∂g∂t = − Ric m,n ( L ) + Kg ) ,∂φ∂t = − R − ∆ φ − |∇ φ | m − n − nK, the potential function f = − log u satisfies the Hessian equation ∇ f = (cid:18) K t (cid:19) g, and moreover ∇ φ · ∇ f = − ( m − n )(1 + Kt )2 t . We can also extend the Hamilton Harnack inequalities to positive solutions to the heatequation ∂ t u = Lu associated with the time dependent Witten Laplacian L = ∆ − ∇ φ · ∇ on compact or complete Riemannian manifolds equipped with a variant of ( − K, m )-superRicci flow. To save the length of the paper, we will do it in a forthcoming paper. See [14].The rest of this paper is organized as follows. In Section 3, we prove Theorem 2.1,Theorem 2.2 and Theorem 2.3. In Section 4, we prove Theorem 2.6 and Theorem 2.7. InSection 5, we compare the W -entropy in Theorem 2.6 and Theorem 2.7 with the W -entropydefined in our previous paper [13].This paper is an improved version of a part of our previous preprint [14]. Due to thelimit of the length of the paper, we split [14] into several papers. See also [15, 16, 17]. By the generalized Bochner-Weitzenb¨ock formula, we have( L − ∂ t ) |∇ u | u = 2 u (cid:12)(cid:12)(cid:12)(cid:12) ∇ u − ∇ u ⊗ ∇ uu (cid:12)(cid:12)(cid:12)(cid:12) + 2 u Ric ( L )( ∇ u, ∇ u ) . L − ∂ t ) |∇ u | u ≥ nu (cid:12)(cid:12)(cid:12)(cid:12) ∆ u − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) + 2 u Ric ( L )( ∇ u, ∇ u ) . Applying the inequality ( a + b ) ≥ a α − b α , ∀ α > , to a = ∂ t u − |∇ u | u , b = ∇ φ · ∇ u , and α = m − nn , we have( L − ∂ t ) |∇ u | u ≥ mu (cid:12)(cid:12)(cid:12)(cid:12) ∂ t u − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) + 2 u Ric m,n ( L )( ∇ u, ∇ u ) . Hence, under the condition
Ric m,n ( L ) ≥ − K , it holds( L − ∂ t ) |∇ u | u ≥ mu (cid:12)(cid:12)(cid:12)(cid:12) ∂ t u − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) − K |∇ u | u . Let h = ∂u∂t − e − Kt |∇ u | u + e Kt m t u. Then lim t → + h ( t ) = + ∞ , and( ∂ t − L ) h ≥ mu e − Kt (cid:12)(cid:12)(cid:12)(cid:12) ∂ t u − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) − e Kt m t u. (18)We now prove that h ≥ M × R + . In compact case, suppose that h attends itsminimum at some ( x , t ) and h ( x , t ) <
0. Then, at ( x , t ), it holds ∂h∂t ≤ , ∆ h ≥ , ∇ h = 0 . Thus at ( x , t ), ( ∂ t − L ) h ≤
0. On the other hand, as h ( x , t ) <
0, we have0 ≤ e Kt m t u < e − Kt |∇ u | u − ∂u∂t ≤ |∇ u | u − ∂u∂t , and hence by (18) we have ( ∂ t − L ) h > . This finishes the proof of Theorem 2.1 in compact case.In complete non-compact case, let f = log u , and let F = te − Kt ( e − Kt |∇ f | − f t ) = te − Kt |∇ f | − te − Kt f t . Obviously, F (0 , x ) ≡
0. We shall prove that F ≤ m . By direct calculation LF = te − Kt L |∇ f | − te − Kt Lf t ∂ t F = (1 − Kt ) e − Kt |∇ f | + (2 Kt − e − Kt f t + te − Kt ∂ t |∇ f | − te − Kt f tt ,
10e have ( L − ∂ t ) F = te − Kt ( L − ∂ t ) |∇ f | − te − Kt ( L − ∂ t ) f t +(4 Kt − e − Kt |∇ f | − (2 Kt − e − Kt f t . By the generalized Bochner formula, it holds( L − ∂ t ) |∇ f | = 2 |∇ f | + 2 Ric ( L )( ∇ f, ∇ f ) − ∇ f ( ∇ f, ∇ f ) . Note that Lf t = L (cid:18) Luu (cid:19) = L uu − h∇ Lu, ∇ uu i + Lu (cid:18) − Luu + 2 |∇ u | u (cid:19) ,∂ t f t = ∂ t (cid:18) Luu (cid:19) = L uu − | Lu | u , which yields ( L − ∂ t ) f t = 2 Lu |∇ u | u − h∇ Lu, ∇ uu i = − ∇ f ( ∇ f, ∇ f ) − h∇ Lf, ∇ f i . Hence ( L − ∂ t ) F = 2 te − Kt [ |∇ f | + 2( e Kt − ∇ f ( ∇ f, ∇ f )]+2 te − Kt Ric ( L )( ∇ f, ∇ f ) + 2 te − Kt h∇ Lf, ∇ f i +(4 Kt − e − Kt |∇ f | − (2 Kt − e − Kt ( Lf + |∇ f | ) . Now F = te − Kt (1 − e Kt ) |∇ f | − te − Kt Lf, h∇ F, ∇ f i = 2 te − Kt (1 − e Kt ) ∇ f ( ∇ f, ∇ f ) − te − Kt h∇ Lf, ∇ f i . Therefore( L − ∂ t ) F = 2 te − Kt |∇ f | − h∇ F, ∇ f i +2 te − Kt (cid:0) Ric ( L )( ∇ f, ∇ f ) + K |∇ f | (cid:1) + (2 Kt − t F. Note that |∇ f | ≥ n | ∆ f | ≥ m | Lf | − m − n ∇ φ ⊗ ∇ φ ( ∇ f, ∇ f ) . Thus( L − ∂ t ) F ≥ te − Kt | Lf | m − h∇ F, ∇ f i +2 te − Kt (cid:0) Ric m,n ( L )( ∇ f, ∇ f ) + K |∇ f | (cid:1) + (2 Kt − t F ≥ te − Kt m (cid:20) ( te − Kt ( e − Kt − |∇ f | − F ) t e − Kt (cid:21) − h∇ F, ∇ f i + (2 Kt − t F ≥ te − Kt ( e − Kt − |∇ f | − F ] mt − h∇ F, ∇ f i + (2 Kt − t F. Similarly to [9], let η be a C -function on [0 , ∞ ) such that η = 1 on [0 ,
1] and η = 0on [2 , ∞ ), with − C η / ( r ) ≤ η ′ ( r ) ≤
0, and η ′′ ( r ) ≥ C , where C > C > ρ ( x ) = d ( o, x ) and define ψ ( x ) = η ( ρ ( x ) /R ). Since ρ is Lipschitz on thecomplement of the cut locus of o , ψ is a Lipschitz function with support in B ( o, R ) × [0 , ∞ ).As explained in Li and Yau [8], an argument of Calabi allows us to apply the maximumprinciple to ψF . Let ( x , t ) ∈ M × [0 , T ] be a point where ψF achieves the maximum.Then, at ( x , t ), ∂ t ( ψF ) ≥ , ∆( ψF ) ≤ , ∇ ( ψF ) = 0 . This yields ( L − ∂ t )( ψF ) = ∆( ψF ) − ∇ φ · ∇ ( ψF ) − ∂ t ( ψF ) ≤ . Similarly to [9], we have( L − ∂ t )( ψF ) = ψ ( L − ∂ t ) F + ( Lψ ) F + 2 ∇ ψ · ∇ F ≥ ψ ( L − ∂ t ) F − A ( R ) F + 2 ∇ ψ · ∇ F ≥ ψ ( L − ∂ t ) F − A ( R ) F + 2 h∇ ψ, ∇ ( ψF ) i ψ − − F |∇ ψ | ψ − . where we use Lψ ≥ − A ( R ) := − C R ( m − √ K coth( √ KR ) − C R , and for some constant C > |∇ ψ | ψ ≤ C R . Let C ( n, K, R ) = C R ( m − √ K coth( √ KR ) + C + C R . At the point ( x , t ), we have0 ≥ ψ ( L − ∂ t ) F − ( A ( R ) + 2 |∇ ψ | ψ − ) F ≥ ψ (cid:20) te − Kt ( e − Kt − |∇ f | − F ] mt − h∇ F, ∇ f i + (2 Kt − t F (cid:21) − C ( n, K, R ) F ≥ ψ mt F + ψ e − Kt (1 − e − Kt ) |∇ f | m F + 2 F h∇ ψ, ∇ f i + (cid:20) (2 K − t ) ψ − C ( n, K, R ) (cid:21) F ≥ ψ mt F + ψ e − Kt (1 − e − Kt ) |∇ f | m F − F |∇ ψ ||∇ f | + (cid:20) (2 K − t ) ψ − C ( n, K, R ) (cid:21) F ≥ ψ mt F + ψ e − Kt (1 − e − Kt ) |∇ f | m F − C R F ψ / |∇ f | + (cid:20) (2 K − t ) ψ − C ( n, K, R ) (cid:21) F. Multiplying by t on both sides, and using the Cauchy-Schwartz inequality, we get0 = ψ m F + tF (cid:20) ψ e − Kt (1 − e − Kt ) |∇ f | m − C R ψ / |∇ f | (cid:21) + [(2 Kt − ψ − C ( n, K, R ) t ] F ≥ ψ m F + (cid:20) (2 Kt − ψ − C ( n, K, R ) t − C mt e − Kt (1 − e − Kt ) R (cid:21) F. Notice that the above calculation is done at the point ( x , t ). Since ψF reaches its maximumat this point, we can assume that ψF ( x , t ) >
0. Thus0 ≥ m ( ψF ) − (cid:20) C ( n, K, R ) t + C m e − Kt (1 − e − Kt ) R t (cid:21) ( ψF ) , which yields that, for any ( x, t ) ∈ B R × [0 , T ], F ( x, t ) ≤ ( ψF )( x , t ) ≤ m (cid:20) C ( n, K, R ) t + C m e − Kt (1 − e − Kt ) R t (cid:21) ≤ m (cid:20) C ( n, K, R ) T + max t ∈ [0 ,T ] C mt e − Kt (1 − e − Kt ) R (cid:21) . R → ∞ , we obtain F ≤ m . The proof of Theorem 2.1 is completed. (cid:3) . The proof is as the same as the one of Corollary 2.2 in [6]. For the completeness wereproduce it as follows. Let l ( x, t ) = log u ( x, t ). Then the Hamilton Harnack inequality isequivalent to ∂l∂t − e − Kt |∇ l | + e Kt m t ≥ . (19)Let γ : [0 , T ] → M be a geodesic with reparametrization by arc length s : [ τ, T ] → [0 , T ] sothat γ ( s ( τ )) = x and γ ( s ( T )) = y . Let S ( t ) = dγ ( s ( t )) dt = ˙ γ ( s ( t )) ˙ s ( t ). Then | ˙ γ ( s ( t )) | = 1.Integrating along γ ( s ( t )) from t = τ to t = T , we have l ( y, T ) − l ( x, τ ) = Z Tτ (cid:20) ∂l∂t + ∇ l · S (cid:21) dt. By the Cauchy-Schwartz inequality e − Kt |∇ l | + 14 e Kt | S | ≥ ∇ l · S From this and (19) we obtain l ( y, T ) − l ( x, τ ) ≥ − Z Tτ e Kt | S | dt − Z Tτ m t e Kt dt. Note that d ( x, y ) = R Tτ | S | dt = R Tτ ds ( t ). Choosing s ( t ) = a [ e − Kτ − e − Kt ], with a = d ( x, y ) e − Kτ − e − KT , we have Z Tτ e Kt | S | dt = Z Tt e Kt ˙ s ( t ) dt = 2 Kd ( x, y ) e − Kτ − e − KT . Therefore l ( y, T ) − l ( x, τ ) ≥ − Z Tt e Kt ˙ s ( t ) dt − Z Tτ m t e Kt dt = − Kd ( x, y )2( e − Kτ − e − KT ) − Z Tτ m t e Kt dt. Note that R Tτ e Kt t dt ≤ log (cid:0) Tτ (cid:1) + e KT − e Kτ . Thuslog u ( y, T ) − log u ( x, τ ) ≥ − Kd ( x, y )2( e − Kτ − e − KT ) − m (cid:20) log (cid:18) Tτ (cid:19) + e KT − e Kτ (cid:21) . Using − e − x ≤ xx , we can derive the desired estimate. (cid:3) .3 Proof of Theorem 2.3 Let ψ ( t ) = − e − Kt K , and h = ψ h Lu + |∇ u | u i − u [ m + 4 log( A/u )]. Then ψ ′ + Kψ = 1 . By (18) , under the assumption
Ric m,n ( L ) ≥ − K we have( ∂ t − L ) |∇ u | u ≤ − mu (cid:12)(cid:12)(cid:12)(cid:12) Lu − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) + 2 K |∇ u | u , which yields( ∂ t − L ) h ≤ − ψmu (cid:12)(cid:12)(cid:12)(cid:12) Lu − |∇ u | u (cid:12)(cid:12)(cid:12)(cid:12) + ψ ′ (cid:20) Lu − |∇ u | u (cid:21) − |∇ u | u . By analogue of Hamilton[6], we can verify that ∂h∂t ≤ Lh whenever h ≥ . Indeed, we can verify this by examining three cases:( i ) If Lu ≤ |∇ u | u , then ( ∂ t − L ) h ≤ ψ ′ ≥ ii ) If |∇ u | u ≤ Lu ≤ |∇ u | u , then ( ∂ t − L ) h ≤ ψ ′ ≤ iii ) If 3 |∇ u | u ≤ Lu , then whenever h ≥
0, we have2 (cid:20) Lu − |∇ u | u (cid:21) ≥ Lu + |∇ u | u = hψ + mu + 4 u log( A/u ) ψ ≥ muψ , which yields, since ψ ′ ≤
1, we have( ∂ t − L ) h ≤ ( ψ ′ − (cid:20) Lu − |∇ u | u (cid:21) − |∇ u | u ≤ . Note that h ≤ t = 0. By the weak maximum principle on complete Riemannianmanifolds, see e.g. Theorem 12.10 in [5], we conclude that h ≤ t ∈ [0 , T ]. Thus Luu + |∇ u | u ≤ K − e − Kt [ m + 4 log( A/u )] . This completes the proof of Theorem 2.3. (cid:3)
The lower bound estimate of (18) follows from (15). It remains to prove the upper boundestimate. Recall the following
Proposition 3.1 ([12]) Suppose that there exist some constants m ≥ n , m ∈ N and K ≥ such that Ric m,n ( L ) ≥ − K . Then, for any small ε > , there exist some constants C i = C i ( m, n, K, ε ) > , i = 1 , , such that for all x, y ∈ M and t > , p t ( x, y ) ≤ C µ ( B y ( √ t )) exp (cid:18) − d ( x, y )4(1 + ε ) t + αεKt (cid:19) × (cid:18) d ( x, y ) + √ t √ t (cid:19) m/ exp p ( m − Kd ( x, y )2 ! , here α is a constant depending only on m , and p t ( x, y ) ≥ C e − (1+ ε ) λ K,m t µ − ( B y ( √ t )) exp (cid:18) − d ( x, y )4(1 − ε ) t (cid:19) " √ Kd ( x, y )sinh √ Kd ( x, y ) m − , where λ K,m = ( m − K . Fix
T >
0, and let u ( t, x ) be a positive and bounded solution to the heat equation ∂ t u = Lu , t ∈ (0 , t ). Let A := sup { u ( t, x ) : 0 ≤ t ≤ t , x ∈ M } . By Hamilton’s Harnack inequality (17), we have t∂ t log u ( x, t ) ≤ (1 + Kt ) [ m + 4 log( A/u ( t, x ))] , ∀ ( t, x ) ∈ [0 , t ] × M. (20)Let s ∈ (0 , T ], y ∈ M , t = s/ u ( t, x ) = p s/ t ( x, y ). By (20) and using the upperbound and lower bound estimates of the heat kernel p t ( x, y ) in Proposition 3.1, we have t ∂ t log p s/ t ( x, y ) ≤ C K,m,T (1 +
Kt/ × " d ( x, y ) √ t + log C C µ ( B ( y, p s/ t )) µ ( B ( y, p s/ (cid:18) C d ( x, y ) s/ t + C d ( x, y ) (cid:19)! . In particular, taking t = s/ s by t we get t t∂ t log p t ( x, y ) ≤ C K,m,T (1 +
Kt/ × " d ( x, y ) √ t + log C C µ ( B ( y, √ t )) µ ( B ( y, p t/ (cid:18) C d ( x, y ) t + C d ( x, y ) (cid:19)! . By the generalized Bishop-Gromov volume comparison theorem for weighted volume mea-sure, see [23, 19, 9, 25], as
Ric m,n ( L ) ≥ − K , for all R > r > y ∈ M , we have µ ( B ( y, R )) µ ( B ( y, r )) ≤ (cid:18) Rr (cid:19) m exp (cid:16)p ( m − KR (cid:17) . It follows that t∂ t log p t ( x, y ) ≤ C K,m,T (cid:18) d ( x, y ) t + d ( x, y ) √ t + d ( x, y ) (cid:19) , which yields ∂ x log p t ( x, y ) ≤ C K,m,T (cid:18) d ( x, y ) t + 1 √ t (cid:19) . This completes the proof of Theorem 2.4. (cid:3) W -entropy for Witten Laplacian with C D ( − K, m ) -condition Recall that, Perelman [22] introduced the W -entropy and proved its monotonicity along theconjugate heat equation associated to the Ricci flow. In [20, 21], Ni proved the monotonicityof the W -entropy for the heat equation of the usual Laplace-Beltrami operator on completeRiemannian manifolds with non-negative Ricci curvature. In [10, 12], the second author ofthis paper proved the W -entropy formula and its monotonicity and rigidity theorems forthe heat equation of the Witten Laplacian on complete Riemannian manifolds satisfyingthe CD (0 , m )-condition and gave a probabilistic interpretation of the W -entropy for theRicci flow. In [13], we gave a new proof of the W -entropy formula obtained in [10] forthe Witten Laplacian by using Ni’s W -entropy formula to the Laplace-Beltrami operatoron M × S m − n equipped with a suitable warped product Riemannian metric, and furtherproved the monotonicity of the W -entropy for the heat equation of the time dependentWitten Laplacian on compact Riemannian manifolds equipped with the super Ricci flowwith respect to the m -dimensional Bakry-Emery Ricci curvature. As we have already seenin Section 1, there is a close connection between the Perelman W -entropy for the heatequation of the Witten Laplacian and the Li-Yau Harnack inequality (10) on completeRiemannian manifolds satisfying the CD (0 , m )-condition. In this section, we will introducethe Perelman W -entropy and prove its monotonicity for the heat equation of the WittenLaplacian on complete Riemannian manifolds with the CD ( − K, m )-condition.Recall the following entropy dissipation formulas for the heat equation of the WittenLaplacian on complete Riemannian manifolds with bounded geometry condition. In thecase of compact Riemannian manifolds, it is a well-known result due to Bakry and Emery[1].
Theorem 4.1 ([10, 12, 13]) Let ( M, g ) be a complete Riemannian manifold with boundedgeometry condition, and φ ∈ C ( M ) such that ∇ k φ are uniformly bounded on M for ≤ k ≤ . Let u be the fundamental solution to the heat equation ∂ t u = Lu . Let H ( u ( t )) = − Z M u log udµ. Then ddt H ( u ( t )) = Z M |∇ log u | udµ, (21) d dt H ( u ( t )) = − Z M Γ ( ∇ log u, ∇ log u ) udµ, (22) where Γ ( ∇ log u, ∇ log u ) = |∇ log u | + Ric ( L )( ∇ log u, ∇ log u ) . Let (
M, g, φ ) be as in Theorem 4.1. Inspired by [22, 20, 10, 12, 13], we define H m,K ( u, t ) = − Z M u log udµ − Φ m,K ( t ) , where Φ m,K ∈ C ((0 , ∞ ) , R ) satisfiesΦ ′ m,K ( t ) = m t e Kt , ∀ t > . The first order entropy dissipation formula (21) holds if φ ∈ C ( M ) such that Ric m,n ( L ) ≥ − K . roposition 4.2 Let ( M, g ) be a complete Riemannian manifold with bounded geometrycondition, φ ∈ C ( M ) be such that ∇ k φ are uniformly bounded on M for ≤ k ≤ . Then,under the condition Ric m,n ( L ) ≥ − K , we have ddt H K,m ( u, t ) ≤ . Proof . By the entropy dissipation formulas in Theorem 4.1 and using the fact R M ∂ t udµ = R M Ludµ = 0, we have ddt H m,K ( u, t ) = Z M (cid:20) |∇ u | u − m t e Kt (cid:21) udµ = Z M (cid:20) |∇ u | u − m t e Kt − e Kt ∂ t uu (cid:21) udµ. By the Hamilton Harnack inequality in Theorem 2.1, we have ddt H m,K ( u, t ) ≤ . (cid:3) Proposition 4.3
Under the same condition as in Theorem 2.6, we have d dt H m,K ( u, t ) = − Z M [ |∇ log u | + Ric ( L )( ∇ log u, ∇ log u )] udµ − (cid:18) mKt − m t (cid:19) e Kt . Proof . Indeed, by the second order dissipation formula of the Boltzmann entropy in Theorem4.1, we have ddt Z M |∇ u | u dµ = − Z M [ |∇ log u | + Ric ( L )( ∇ log u, ∇ log u )] udµ. Combining this with (23), Proposition 4.3 follows. (cid:3)
Based on the Hamilton differential Harnack inequality (15) in Theorem 2.1, we nowintroduce the W -entropy for the heat equation (9) of the Witten Laplacian on completeRiemannian manifolds with the CD ( − K, m )-condition as follows W m,K ( u, t ) = ddt ( tH m,K ( u, t )) . By the entropy dissipation formula in Theorem 4.1, we have W m,K ( u, t ) = Z M (cid:2) t ( |∇ log u | − Φ ′ m,K ( t )) − log u − Φ m,K ( t ) (cid:3) udµ = Z M (cid:2) t (2 L ( − log u ) − |∇ log u | ) − log u − Φ m,K ( t ) − Φ ′ m,K ( t ) (cid:3) udµ. We are now in a position to prove the main result of this section, i.e., Theorem 2.6.
Proof of Theorem 2.6 . By (23) and Proposition 4.3, we have ddt W m,K ( u, t ) = − t (cid:20)Z M [ |∇ log u | + Ric ( L )( ∇ log u, ∇ log u )] udµ + (cid:18) mKt − m t (cid:19) e Kt (cid:21) +2 Z M (cid:20) |∇ u | u − m t e Kt (cid:21) udµ. (cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) e Kt t + a ( t ) (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) = |∇ log u | + 2 (cid:18) e Kt t + a ( t ) (cid:19) ∆ log u + n (cid:18) e Kt t + a ( t ) (cid:19) . By direct calculation, we have ddt W m,K ( u, t ) = − t Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) e Kt t + a ( t ) (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) udµ − t Z M (cid:18) Ric m,n ( L ) + (cid:18) a ( t ) − − e Kt t (cid:19) g (cid:19) ( ∇ log u, ∇ log u ) udµ +2 nt (cid:18) e Kt t + a ( t ) (cid:19) − me Kt t − mKe Kt +2( e Kt + 2 ta ( t )) Z M ∇ φ · ∇ log u udµ − t Z M |∇ φ · ∇ log u | m − n udµ. Let a ( t ) be chosen such that 2 a ( t ) − − e Kt t = K . Then ddt W m,K ( u, t ) = − t Z M "(cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) K t (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) + ( Ric m,n ( L ) + Kg )( ∇ log u, ∇ log u ) udµ +2 nt (cid:18) t + K (cid:19) − me Kt t − mKe Kt +2(1 + Kt ) Z M ∇ φ · ∇ log u udµ − t Z M |∇ φ · ∇ log u | m − n udµ. Combining this with1 m − n Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ φ · ∇ log u − ( m − n )(1 + Kt )2 t (cid:12)(cid:12)(cid:12)(cid:12) udµ = ( m − n )(1 + Kt ) t − Ktt Z M ∇ φ · ∇ log u udµ + Z M |∇ φ · ∇ log u | m − n udµ, and noting that 2 nt (cid:18) t + K (cid:19) − me Kt t − mKe Kt + ( m − n )(1 + Kt ) t = m t (cid:2) (1 + Kt ) − e Kt (1 + 4 Kt ) (cid:3) , we can derive the desired W -entropy formula. The rest of the proof is obvious. (cid:3) In particular, taking m = n , φ ≡ g is a fixed Riemannian metric, we have thefollowing W -entropy formula for the heat equation of the Laplace-Beltrami operator onRiemannian manifolds, which extends Ni’s result in [20] for K = 0. Theorem 4.4
Let ( M, g ) be a complete Riemannian manifold with bounded geometry con-dition. Let u be the fundamental solution to the heat equation ∂ t u = ∆ u . Then ddt W n,K ( u, t ) = − t Z M "(cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) K t (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) + ( Ric + Kg )( ∇ log u, ∇ log u ) udµ − n t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . n particular, if Ric ≥ − K , then, for all t ≥ , we have ddt W n,K ( u, t ) ≤ − n t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . Moreover, the equality holds at some time t = t > if and only if M is an Einsteinmanifold, i.e., Ric = − Kg , and the potential function f = − log u satisfies the shrinkingsoliton equation, i.e., Ric + 2 ∇ f = gt . By analogue of the W -entropy for the heat equation of the Witten Laplacian on com-plete Riemannian manifolds with the CD ( − K, m )-condition, we can prove the W -entropyformula for the heat equation of the time dependent Witten Laplacian on compact manifoldsequipped with a ( − K, m )-super Ricci flow. To do so, let us recall the entropy dissipationformula on compact manifolds with time dependent metrics and potentials.
Theorem 4.5 ([13]) Let ( M, g ( t ) , t ∈ [0 , T ]) be a family of compact Riemannian manifoldswith potential functions φ ( t ) ∈ C ∞ ( M ) , t ∈ [0 , T ] . Suppose that g ( t ) and φ ( t ) satisfy theconjugate equation ∂φ∂t = 12 Tr (cid:18) ∂g∂t (cid:19) . Let L = ∆ g ( t ) − ∇ g ( t ) φ ( t ) · ∇ g ( t ) be the time dependent Witten Laplacian on ( M, g ( t ) , φ ( t )) . Let u be a positive solution ofthe heat equation ∂ t u = Lu with initial data u (0) satisfying R M u (0) dµ (0) = 1 . Let H ( u, t ) = − Z M u log udµ be the Boltzmann-Shannon entropy for the heat equation ∂ t u = Lu . Then ∂∂t H ( u, t ) = Z M |∇ log u | g ( t ) udµ,∂ ∂t H ( u, t ) = − Z M (cid:20) |∇ log u | + (cid:18) ∂g∂t + Ric ( L ) (cid:19) ( ∇ log u, ∇ log u ) (cid:21) udµ. Proof of Theorem 2.7 . Base on the entropy dissipation formulas in Theorem 4.5, theproof of Theorem 2.7 is similar to the one of Theorem 2.6. See [13] for the case K = 0. (cid:3) W -entropy functional To end this paper, let us mention that in our previous paper [13] we introduced another W -entropy functional for the heat equation associated with the Witten Laplacian on completeRiemannian manifolds satisfying the CD ( − K, m )-condition as follows f W m,K ( u ) = ddt ( t e H m,K ( u )) , e H m,K ( u ) = − Z M u log udµ − (cid:20) m t (1 + log(4 πt )) + mKt Kt ) (cid:21) , and we proved that ddt f W m,K ( u ) = − t Z M "(cid:12)(cid:12)(cid:12)(cid:12) ∇ log u + (cid:18) K t (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) + ( Ric m,n ( L ) + Kg )( ∇ log u, ∇ log u ) udµ − tm − n Z M (cid:12)(cid:12)(cid:12)(cid:12) ∇ φ · ∇ log u − ( m − n )(1 + Kt )2 t (cid:12)(cid:12)(cid:12)(cid:12) udµ. It is interesting to compare the W -entropy defined in [13] with the W -entropy definedin this paper, and to compare the W -entropy formula proved in [13] with the W -entropyformula obtained in Theorem 2.6. Indeed, lettingΨ m,K ( t ) = Φ m,K ( t ) − (cid:20) m t (1 + log(4 πt )) + mKt Kt ) (cid:21) , we have f W m,K ( u ) − W m,K ( u ) = ddt ( t Ψ m,K ( t )) , Moreover, by direct calculation we have ddt ( f W m,K ( u ) − W m,K ( u )) = d dt ( t Ψ m,K ( t ))= m t (cid:2) e Kt (1 + 4 Kt ) − (1 + Kt ) (cid:3) . This explains clearly the difference between the W -entropy defined in [13] and the W -entropydefined in this paper, and the difference between the W -entropy formula proved in [13] withthe W -entropy formula obtained in Theorem 2.6.Similarly, we can reformulate Theorem 2.7 in terms of f W m,K . See [13, 17]. Acknowledgement . Part of this work was done when the second author visited the Institutdes Hautes Etudes Scientifiques and the Max-Planck Institute for Mathematics Bonn. Theauthors would like to thank Professors D. Bakry, J.-M. Bismut, M. Ledoux, N. Mok, K.-T.Sturm, A. Thalmaier, F.-Y. Wang and Dr. Yuzhao Wang for their interests and helpfuldiscussions during the preparation of this paper. We are very grateful to anonymous refereefor his careful reading and for his very nice comments which lead us to improve the writtingof this paper.
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