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Scalar curvature, entropy, and generalized Ricci flow

We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles.


Introduction
A dominant theme in the analysis of Ricci flow is the understanding of curvature positivity conditions preserved by the flow [13,14,22,23,25]. Most fundamental among these, as observed by Hamilton in his original paper [22], is the preservation of a lower bound on the scalar curvature. This bound is essential for detailed analyses of heat kernels, ancient solutions, and singularity formation of Ricci flow (cf. e.g. [11,16,18,26,47,48,49]). A second dominant theme is the key role played by self-similar solutions of the flow, i.e. Ricci solitons, which partly indicate the subtle interplay between Ricci flow and the diffeomorphism group. Such solutions, and the interaction between Ricci flow and the diffeomorphism group, lie at the foundation of various key estimates for Ricci flow, such as Hamilton's Harnack estimate [24], and Perelman's energy, entropy, reduced volume functionals, and differential Harnack estimate [32]. These various tools combine to reveal the structure of singular sets of Ricci flow [32,7,8,9], leading to deep topological applications [33,10]. In this paper we extend the fundamental circle of ideas around scalar curvature monotonicity, Harnack estimates, and heat kernels to generalized Ricci flow. In the process we obtain several new estimates for Ricci flow, in particular a new family of convex Nash entropies, and pseudolocality estimates Given a smooth manifold M, a one-parameter family of metrics g t and closed three-forms H t is a solution of generalized Ricci flow if where H 2 (X, Y ) = i X H, i Y H . This equation arises independently in mathematical physics [31,34], complex geometry [40,42], and generalized geometry [19,36,41], and we refer to [20] for further background. Some global existence and convergence results can be found in [2,27,39,37]. Note that H ≡ 0 is preserved by the flow (cf. [20] Proposition 4.20), and the metric then solves Ricci flow. Thus in the remainder of this paper many results are for generalized Ricci flow, with the attendant results for Ricci flow occurring as a special case.
1.1. Scalar curvature monotonicity. Given a metric g, closed three-form H, and smooth function f , let Rc H,f = Rc − 1 4 H 2 + ∇ 2 f − 1 2 d * g H + i ∇f H , R H,f := R − 1 12 |H| 2 + 2∆f − |∇f | 2 . The tensor Rc H,f reduces to the Ricci tensor of the Bismut connection with torsion H when f = 0, and in general can be motivated by extending ideas from Bakry-Emery [5] to the Laplacian of the Bismut connection acting on one-forms. The scalar curvature R H,f arises in the Lichnerowicz-type formula for the cubic Dirac operator of Bismut [12] in the case f = 0. The general case occurs when computing this formula using a weighted volume form (cf. [6,32]).
Perelman's energy and entropy monotonicity formulas can be interpreted as differential inequalities for the weighted scalar curvature R f . The key point is to allow the weight u = e −f to evolve by the conjugate heat equation. In this circle of ideas the function u = e −f is the "dilaton" in physics terminology. Interestingly, in mathematical physics literature a different equation is suggested for the dilaton in the RG flow of the H-twisted nonlinear sigma model [34]. Specifically, given (g t , H t ) a solution of generalized Ricci flow, we let = ∂ ∂t − ∆ denote the forward heat operator, and fix φ a solution to the associated dilaton flow, φ = 1 6 |H| 2 . With the setup above we then obtain the following evolution equation for the generalized scalar curvature: , and obtain a monotone curvature quantity (cf. §2.2). Furthermore the tensor H may be replaced with a formal linear combination of differential forms of all degrees, and with an appropriately weighted scalar curvature and dilaton flow one again obtains a monotone curvature quantity (cf. §2.3). One special case of this is the extended Ricci flow system of List [29], coupling to an exact 1-form. Another special case is the Ricci-Yang-Mills flow ( [45,46,35,39]), a coupling of the Ricci and Yang-Mills flows which in the case of abelian structure group corresponds to the case that H is a two-form, specifically the principal curvature.
1.2. Entropy formulas and Harnack estimates. Further structure is revealed when we treat e −φ as a volume density, mirroring the role played by u = e −f in Perelman's work. Perelman's idea underlying his energy and entropy formulas is to let u = e −f be a solution of the conjugate heat equation Given this, one obtains the differential equation This is the pointwise computation underlying the monotonicity of Perelman's F -functional, and including a further weighting of u by a time scale yields Perelman's entropy density monotonicity. There is a curious duality between equations (1. When integrated against e −f dV g , this pointwise formula yields the gradient flow interpretation for generalized Ricci flow. As natural as this differential equation may seem, it is difficult to exploit in part because the conjugate heat operator itself is difficult to control. In the setting of Ricci flow the a priori lower bound on the scalar curvature controls the reaction term in the conjugate heat operator, and this plays a key role in many applications. We get a better behaved heat kernel by including a further weight given by a solution to the dilaton flow. Indeed the conjugate of the heat operator, taken with respect to the measure dm = e −φ dV g , is * We then obtain a generalization of (1. By including further weighting of u by a time scale we also obtain a generalization of Perelman's shrinker entropy monotonicity and differential Harnack estimate, parameterized by the choices of f and φ. In particular, recall Perelman's entropy density To obtain a monotone quantity under generalized Ricci flow we must include a further functional parameter which measures the concentration of H, namely a solution to the conjugate dilaton flow. In particular we let ψ denote a solution of * φ (ψu) = − 1 6 |H| 2 u. We then obtain: This yields that if u approaches a weighted Dirac delta at some forward time then (W H,f +φ + ψ)u 0, generalizing Perelman's Harnack estimate (cf. Corollary 3.14). Due to the presence of the conjugate dilaton flow solution, finding further geometric applications requires a more detailed understanding of the torsion H.

Further applications to Ricci flow.
To conclude we observe some formal extensions of some key results in the analysis of Ricci flow to the setting of weighted scalar curvature. First we note an extension of the definition of the Nash and Perelman entropies, adapted to weighted scalar curvature. In Proposition 4.2 we show convexity of the Nash entropy, extending fundamental observations in [26]. Going further, recall a key application of Perelman's differential Harnack estimate is the pseudolocality estimate [32], which roughly says that almost Euclidean regions will regularize for a short time. The strength of this result is that 'almost Euclidean,' is measured in a very weak sense, namely by a lower scalar curvature bound and an almost-Euclidean isoperimetric inequality. Based on the generalized entropy monotonicity formulas above, we give an extension of this result, involving the weighted scalar curvature and isoperimetric inequality (Theorem 1.7). The proof follows Perelman's original proof until the final stages, where the entropy integrand is manipulated to exploit the weighted scalar curvature bound. In the end we require a technical result relating the weighted isoperimetric inequality to the weighted log-Sobolev inequality (Theorem 4.7), proved using the method of Steiner symmetrization.
Acknowledgements: This work emerged from work with V. Apostolov and Y. Ustinovskiy on scalar curvature in generalized Kähler geometry. Companion work will appear on the role of scalar curvature in generalized Kähler geometry [3], and on the generalized Kähler Calabi-Yau conjecture [1]. We further thank Ben Chow and Richard Bamler for helpful comments.

Weighted scalar curvature monotonicity formulas
In this section we show several monotonicity formulas for weighted scalar curvatures and certain further generalizations. First we prove Proposition 1.1 of the introduction, and derive a rigidity result using the strong maximum principle. Then we extend to a more general setting where df is replaced by an arbitrary 1-form, and then to the setting where the metric flow is coupled to the heat flow for differential forms of arbitrary degree.
2.1. Weighted scalar curvature monotonicity and rigidity results. To begin we formalize some definitions from the introduction.
This data also determines a generalized scalar curvature R H,φ := R − 1 12 |H| 2 + 2∆φ − |∇φ| 2 . Definition 2.2. Given (M, g t , H t ) a solution to generalized Ricci flow, a one-parameter family φ t satisfies the dilaton flow if φ = 1 6 |H| 2 . Remark 2.3. We separate the terminology of a solution to the dilaton flow associated to a solution of generalized Ricci flow to emphasize that the two flows are decoupled, and for instance the initial data for φ is arbitrary. On the other hand, for convenience we will also refer to a triple (g t , H t , φ t ) as a solution of generalized Ricci flow, where a particular solution φ to the dilaton flow has been selected.
Proof. We compute the heat operator acting on each term of R H,φ separately. First, a standard computation (cf. [20] Lemma 5.11) yields Next we have Furthermore using the Bochner formula Combining the above formulas and using the definition of R H,φ yields We further recall the identity (cf. [20] Plugging these into (2.1) yields as claimed.
Proof. This follows from the maximum principle applied to Proposition 2.4 Proof. This follows from the strong maximum principle applied to the evolution equation of Proposition 2.4.

2.2.
One-form scalar curvature monotonicity. In the context of generalized geometry, the dilaton φ, or more accurately its differential dφ, plays the role of a divergence operator [19,20], which is necessary to define the generalized Ricci tensor. A natural class of divergence operators are defined by a 1-form which need not even be closed. Next we extend the results above to this more general setting.
Proof. The proof is nearly identical to that of Proposition 2.4. We first note

Furthermore using the Bochner formula
Using these and arguing as in Proposition 2.4 we obtain as claimed.
Remark 2.10. An a priori lower bound for R H,α as in Corollary 2.5 and a rigidity result as in Corollary 1.3 are immediate consequences.
2.3. Further generalizations of Ricci flow. The results of this paper extend to a more general class of geometric evolution equations coupled to differential forms. In particular, let H = n k=1 H k denote a closed section of Λ * T * M, where the subscript indicates the degree of the differential form, and consider the system of equations where as before H 2 (X, Y ) = i X H, i Y H . We note that the constant factor of 1 2 on the term H 2 may seem arbitrary, but as the coupled PDE for H is linear, this constant can be tuned to any positive value, independently for each value of k if desired. This system of equations obeys the same fundamental regularity of properties as generalized Ricci flow, and has an interpretation as Ricci flow on more general Courant algebroids (cf. [20]). For this setup we define the generalized scalar curvature Also we attach a dilaton flow of the form

is a solution of (2.3) and φ t is a solution of (2.4). Then
Proof. For a closed differential form H k of degree k one has the Bianchi identities Using these a straightforward modification of the proof of Proposition 2.4 gives the result.
Remark 2.12. An a priori lower bound for R H,φ as in Corollary 2.5 and a rigidity result as in Corollary 1.3 are immediate consequences.

Weighted energy and entropy formulas
In [32] Perelman introduced differential inequalities for the weighted scalar curvature, coupled to a solution of the conjugate heat equation. These key estimates complement the a priori scalar curvature lower bound, and underpin the proofs of κ-noncollapsing and pseudolocality for Ricci flow. In this section, complementary to our a priori lower bound for the generalized scalar curvature, we generalize these estimates to the case of generalized Ricci flow, by further coupling to a solution of the dilaton flow. The key point is to treat the auxiliary solution to the dilaton flow as a shift in Perelman's dilaton f , in particular constructing f as a solution to the φ-weighted conjugate heat equation. This leads to a family of differential inequalities with now two auxiliary functional parameters φ and f .
Proof. It follows from Proposition 1.5 that R f +φ is a subsolution of a backwards heat-type equation, therefore by the maximum principle its supremum is nonincreasing as a function of −t, therefore nondecreasing as a function of t.
3.3. Gradient property revisited and steady solitons. The results in the previous subsection underpin a generalization of the gradient flow interpretation of generalized Ricci flow. To begin we recall the usual gradient flow interpretation. Define Following [32], it was shown in [31] it is shown that generalized Ricci flow is the gradient flow of λ. Furthermore, there is a general monotonicity formula for F once u = e −f is imposed to solve the conjugate heat equation along the flow. By explicitly including the dilaton shift φ we get an infinite dimensional family of eigenvalues λ for which generalized Ricci flow is the gradient flow, following Proposition 1.5. Critical points for λ are steady generalized Ricci solitons, and recent constructions and classification results for these objects have appeared in [4,38,44,43], including examples on compact manifolds. The next proposition shows that, on such a steady soliton, the dilaton flow (suitably normalized) converges to the relevant soliton function f . Proof. We recall the basic identities for a soliton: The first follows by tracing Rc H,f = 0 and the second follows by first observing that R f is constant by a Bianchi identity (cf. [20] Proposition 4.33), then observing by integration against f , suitably normalized, that this constant must be λ(g, H). Using these identities we compute As + ∇f is a strictly parabolic operator with no constant term, it follows from standard results that φ t exists for all time, and moreover that φ − f approaches a constant.

Entropy density monotonicity.
Definition 3.9. Given a smooth manifold M, a triple (g, H, F ) of a Riemannian metric, closed three-form H, function F and τ > 0 determine a generalized entropy density In the setting of Ricci flow the quantity W F is Perelman's entropy density, which satisfies a key monotonicity property. To obtain a monotone entropy quantity for generalized Ricci flow we require a further functional parameter, which can be used to measure the concentration of H at a given point.
Proof. Given a smooth function ψ we formally compute * Using that u is positive, the result follows. Proof. First observe, using * φ u = 0, (3.1) Now we compute using Proposition 1.5, We then obtain * and the result follows.

Perelman's Harnack estimate.
To prove the generalization of Perelman's Harnack estimate we require some technical estimates for the conjugate heat kernel along solutions to generalized Ricci flow, which follow from straightforward modifications of results in [15] (cf. also [30,47]). We fix (M n , g t , H t , φ t ) a solution of generalized Ricci flow on M × [0, T ], and let Z(x, t, y, s) denote the fundamental solution of u = 0. We fix p ∈ M and define u(x, t) = Z(p, T, x, t). It follows that u is a solution of the conjugate heat equation * u = 0. We define f by u = (4πτ ) − n 2 e −f as above, where τ = T − t. Furthermore, we fix 0 < t 0 < T and let h t 0 0 denote a smooth compactly supported function, and let h t denote the solution to h = 0 on [t 0 , T ] with initial condition h t 0 . We then claim the following fundamental characteristics for the entropy density W H,f . Proposition 3.13. (cf. [15] Theorem 7.1) Given the setup above, (1) For any t 0 < t < T one has that hW H,f ∈ L 1 (M, g t ).
(2) For any t 0 < t 1 < t 2 < T one has However we obtain a contradiction since by construction M hV e −φ dV gt 0 > 0 and by Proposition 3.13 and the fact that ψ T ≡ 0, lim t→T M hV e −φ dV g = 0.

Weighted Nash entropy and pseudolocality estimates
In this section we give an extension of the Nash entropy convexity for Ricci flow to the setting of weighted scalar curvature. Then we derive a pseudolocality principle for Ricci flow in terms of weighted scalar curvature generalizing Perelman's result [32]. The key new technical point is a weighted isoperimetric inequality and its relationship to a weighted log-Sobolev inequality.
Proof. Since * φ u = 0 an elementary computation shows that we have that Furthermore, using the evolution equation (3.1) satisfied by f and integration by parts we compute Thus, using that M dν ≡ 1, we obtain as claimed. Furthermore we note that we can express Then it follows by Lemma 3.2 and Proposition 3.12 with ψ ≡ 0 that as claimed.
We will show (4.5), dropping the tildes from the notation for simplicity. By an approximation argument it suffices to consider the case ψ 0. For s > 0 let M s = {x ∈ M | ψ s}. Let Γ s = ∂M s , and let F (s) := Vol e −φ dVg (M s ).
Choose r 0 < ∞ such that Vol e −φ dVg ({ψ > 0}) = ω n r n 0 = Vol dV R n (B r 0 ). Choose the unique rotationally invariant function h : R n → R such that Vol dV R n ({h s}) = F (s), and such that h(x) = 0 for |x| r 0 . Furthermore set M ′ s = {h s}, and Γ ′ s = ∂M ′ s . We recall the statement of the co-area formula,

HdAds.
Applying this with H = e −φ |∇ψ| −1 and f = ψ on M t yields This implies that for almost all s one has Thus for an arbitrary Lipschitz function λ we have Integrating by parts we have By construction, we may argue similary with the function λ(h) on R n to obtain Choosing λ(s) = (log s 2 )s 2 and λ(s) = s 2 thus implies This reduces the result to comparing to comparing the gradient terms. By the co-area formula we have   Then |Rm| (p, t) < αt −1 + (ǫr 0 ) −2 whenever 0 < t (ǫr 0 ) 2 and d t (p, p 0 ) ǫr 0 . Remark 4.9. In the hypotheses above there is both a lower bound for φ 0 in item (2) and an implicit upper bound for φ 0 in item (3) (cf. Remark 4.6).
Proof. We follow the proof in [28] and briefly indicate the initial phases. By scale invariance of the hypotheses and conclusion it suffices to consider the case r 0 = 1 and also α < 1 100n . If the theorem is false, there exist sequences ǫ k → 0, δ k → 0 pointed Ricci flow solutions (M k , (p 0,k , 0), g k ) and functions φ k 0 which satisfy the hypotheses of the theorem but for which there exists (p k , t k ) with 0 < t k ǫ k , d(p k , t k ) ǫ k , but |Rm| (p k , t k ) αt −1 k + ǫ −2 k . Choose A k = 1 100nǫ k , and employ ([28] Lemma 31.1) to obtain a new sequence (p k , t k ) which lie at the center of parabolic balls of a controlled size. We let u k = (4π(t k −t)) − n 2 e −F k satisfy * u k = 0 and lim t→t − k u(p, t) = δ p k (p). Furthermore let W k = W F k . It follows from ([28] Lemma 33.4) that there exists β > 0 so that for all sufficiently large k, there exists t k ∈ [t k − 1 2 αQ −1 k , t k ] with B k W k dV k −β, where Q k = |Rm| (p k , t k ), and B k = B t k (p k , t k − t k ).
The aim is to use this to contradict the weighted log-Sobolev inequality at the initial time. We drop the subscripts and work with a particular large k. One constructs a function h on spacetime from a modified distance function satisfying certain estimates we will use below (cf. [28] §34). Using Perelman's Harnack estimate and estimates for h we produce a function u = (4πt) − n 2 e −F at time zero with mass arbitrarily close to 1 which satisfies (cf. [28] (34.6)) We drop the subscript on φ 0 and define f via F = f + φ.  For ǫ, δ sufficiently small it follows then that For δ chosen sufficiently small this contradicts hypothesis (3) using Theorem 4.7.
Remark 4.10. Theorem 1.7 implies via smoothing a diffeomorphism finiteness result for the class of manifolds with a weighted scalar curvature lower bound, almost nonnegative weight, volume upper bound, and almost-Euclidean weighted isoperimetric inequality (cf. [28] Theorem 37.1).