Spinors and mass on weighted manifolds

This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.


Introduction
Manifolds with density, or weighted manifolds, have long appeared in mathematics. A weighted manifold is a Riemannian manifold (M, g) endowed with a function f : M → R, defining the measure e −f dV g . After being introduced by Lichnerowicz in [Lic1,Lic2], more recent attention has been given to the differential geometry of weighted manifolds, including a generalization of Ricci curvature. A central idea of Perelman's spectacular proofs [P] required considering manifolds with density and their evolution. This led him to introduce a notion of weighted scalar curvature which is not the trace of the weighted Ricci curvature of Bakry-Émery. Sometimes called the P-scalar curvature, this weighted scalar curvature has only been moderately studied; see for instance [Fa, AC, LM, D, BH]. This paper shows that the intimate relationship between scalar curvature and the Dirac operator generalizes naturally to the weighted scalar curvature and an associated weighted Dirac operator, defined below. Well-known theorems relating scalar curvature and the Dirac operator include Friedrich's eigenvalue estimate [Fr1], Witten's proof of the positive mass theorem [W1], Gromov-Lawson's obstructions to positive scalar curvature [GL], and the Seiberg-Witten theory [W3]. Here, the first two of said theorems are generalized and then applied to the Ricci flow.
Aside from their applications in Ricci flow, weighted manifolds have proven extremely useful in the context of diffusion operators in analysis and probability theory, starting with Bakry and Émery's celebrated article [BE]. In a more classical Riemannian geometry context, Cheeger-Colding showed that limits of collapsing manifolds are naturally endowed with densities. Such densities differ from those defined by the Riemannian volume form, and the natural object of study is a metric measure space. See also the many extensions to the theory of (R)CD spaces started in [LV, S].
In physics, manifolds with density appear in a number of theories arising from Kaluza-Klein compactifications, via the mechanism of dimensional reduction. The closest to the purpose of this paper is probably Brans-Dicke theory, which motivates the study of manifolds with density and Bakry-Émery's notion of (weighted) Ricci curvature in [GW, WW, LMO]. Also, the weighted version of the Hilbert-Einstein action, introduced by Perelman, appears as the Lagrangian in several gravitational theories; this fact was noted in [CCD+], for instance. Table 1 gives a summary comparison between classical and weighted quantities. The weighted quantities are typically better behaved than their Riemannian counterparts as one can choose a geometrically meaningful density. This idea can be seen as the core of Perelman's proofs [P]. In the context of scalar curvature and mass questions, proofs often employ a conformal change of the metric to reach constant scalar curvature, significantly changing the geometry; see [CP] for a survey of this technique. In contrast, on weighted manifolds, the idea is rather to fix the background geometry while varying the weight in order to obtain a metric with constant weighted scalar curvature.

Riemannian
with density 0.1. Weighted Dirac operator. Section 1 extends classical spin geometry theory to weighted manifolds. The new mathematical object introduced in this section is the weighted Dirac operator, The ∇f term acts by Clifford multiplication, and D denotes the standard (unweighted) Dirac operator. The weighted Dirac operator is self-adjoint with respect to the weighted L 2 -inner product and is unitarily equivalent to the standard Dirac operator; see Proposition 1.20. Differential operators naturally associated with weighted measures have proven invaluable in analysis and geometry. Of particular note is the weighted Laplacian, ∆ f = ∆ − ∇ ∇f , also called the drift Laplacian, f -Laplacian, or Witten Laplacian. When f = |x| 2 4 on R n , then ∆ f is the Ornstein-Uhlenbeck operator. Weighted Laplace operators have been used in Ricci and mean curvature flow to analyze solitons [CM, CZ, MW], and by Witten in his study of Morse theory [W2], for example. Proposition 1.8 proves a weighted Lichnerowicz formula involving the weighted scalar curvature, Proposition 1.15 proves a weighted Ricci identity involving the Bakry-Émery Ricci curvature, Theorem 1.23 generalizes the classical lower bound for Dirac eigenvalues to the weighted setting: on a closed, weighted spin manifold, any eigenvalue λ of D f satisfies Furthermore, the same lower bound also holds for eigenvalues of the standard Dirac operator. Forthcoming work will study weighted spin manifolds with boundary [BO2].
0.2. Weighted asymptotically Euclidean manifolds. A fundamental quantity associated with an asymptotically Euclidean (AE) manifold (M n , g) is the ADM mass [ADM], denoted m(g). Section 2 introduces a quantity extending the ADM mass to the weighted setting: the weighted mass of an AE manifold with weight function f is defined as where S ρ is a coordinate sphere of radius ρ with outward normal ν and area form dA. The normalization for m used in this paper is related to Bartnik's [B] by m = c n m ADM , where c n = 2(n−1)ω n−1 and ω n−1 is the area of the unit sphere in R n ; this simplifies the formulas to follow. Theorem 2.5 shows that the weighted mass of a spin manifold satisfies a weighted Witten formula: if the weighted scalar curvature is nonnegative and f decays suitably rapidly at infinity, there exists an asymptotically constant weighted-harmonic spinor ψ of norm 1 at infinity and satisfying Moreover, Theorem 2.13 proves a positive weighted mass theorem on spin manifolds: if the weighted scalar curvature is nonnegative and f decays suitably rapidly at infinity, then (0.7) m f (g) ≥ 0, with equality iff (M n , g) ∼ = (R n , g euc ) andˆR n (∆ f f ) e −f dV geuc = 0.
By way of a parenthetical remark: using work of Nakajima [N] (see [DO1]), the results of this section have straightforward extensions to asymptotically locally Euclidean spaces of dimension 4 with subgroup SU(2) at infinity, though they are not pursued in this paper. 0.2.1. Weighted mass and Ricci flow. ADM mass does not measure how far a manifold is from the Euclidean metric, except in an asymptotic way at infinity. Indeed, one striking way to see this is that 3-dimensional Ricci flow (with surgery) starting at an AE metric with nonnegative scalar curvature converges to Euclidean space [Li]; however, mass is constant along the flow and thus does not detect the improvement of the geometry [DM,OW,Ha2,Li].
On the other hand, with a suitable choice of weight function f , the weighted mass indeed measures how far an AE manifold is from Euclidean space: the most natural choice for f is the unique f g decaying at infinity and solving R fg ≡ 0. Theorem 2.17 shows that such an f g exists on any AE manifold with nonnegative scalar curvature. This surprisingly yields the formula where λ ALE (g) is the renormalized Perelman functional introduced by Deruelle and the second author [DO1]. Equality (0.8) is the content of Theorem 2.17, and is unexpected at first sight since λ ALE stems from a variational principle on the whole manifold, and a priori is not a boundary term.
(The notation for λ ALE is adopted from [DO1], since the results here also apply to ALE spaces.) The renormalized Perelman functional is the correct modification of Perelman's λ-functional (for closed manifolds) to AE manifolds: it has the crucial property that Ricci flow, ∂ t g = −2Ric, is its gradient flow [DO1,Ha1]. Thus equality (0.8) implies that a Ricci flow on an AE manifold with nonnegative scalar curvature is the gradient flow of the weighted mass (see Corollary 2.20): (0.9) d dt m fg (g) = −2ˆM |Ric + Hess fg | 2 e −fg dV ≤ 0, and equality implies Ricci-flatness. Together, (0.6), (0.8), and (0.9) imply the following monotonicity formula along Ricci flow for the weighted spinorial Dirichlet energy of a weighted Witten spinor: This monotonicity formula stands in contrast to the constancy of ADM mass along Ricci flow, which implies that for an (unweighted) Witten spinor ϕ, the integral´M (|∇ϕ| 2 + 1 4 R|ϕ| 2 )dV is constant along Ricci flow. Further applications of spin geometry to the Ricci flow, including a direct proof of (0.10) via the first variation, will be presented in forthcoming work [BO1].
Equality (0.8) additionally implies that all of the advantages of λ ALE over the ADM mass also hold for the weighted mass. In addition to those already stated, the key advantages of the weighted mass over the ADM mass are as follows: like ADM mass, m fg (g) is nonnegative on any spin AE manifold, and vanishes only on Euclidean space; m fg (g) satisfies a Łojasiewicz inequality measuring the distance to Euclidean space; m fg (g) is real-analytic on weighted Hölder spaces, where neither mass, nor the L 1 -norm of scalar curvature are defined; even when an AE manifold has some negative scalar curvature, m fg (g) is nonnegative and detects how far from Euclidean space the geometry is, allowing for stability analysis of gravitational instantons under general perturbations [DO2].

Weighted Dirac operator
Let (M n , g) be a complete Riemannian spin n-manifold without boundary. The spin bundle ΣM → M is a complex vector bundle of rank 2 ⌊ n 2 ⌋ , equipped with a Hermitian metric, Clifford multiplication, and connection. These objects satisfy compatibility conditions which are stated below. A spinor field, or simply spinor, is a section of the bundle ΣM . For background on spin geometry, see the book [BHM+], whose notation and conventions are adopted here. Let where D = e i · ∇ i is the standard (Atiyah-Singer) Dirac operator and · denotes Clifford multiplication. (Throughout this paper, 1-forms and vector fields will often be identified without explicit mention.) The weighted Dirac operator is the Dirac operator associated with the modified spin connection where ∇ is the standard spin connection induced by the Levi-Civita connection. The modified spin connection ∇ f is not metric compatible with the standard metric [BHM+,Prop. 2.5] on the spin bundle, ·, · , however, it is compatible with the modified metric ·, · f := ·, · e −f , that is any vector field X and spinors ψ, ϕ. Moreover, since Clifford multiplication is parallel with respect to the standard spin connection, it is also parallel with respect to ∇ f . This means that for any vector fields X, Y and spinor ψ.
The weighted Dirac operator satisfies the following weighted integration by parts formula on and hence is self-adjoint on W 1,2 (e −f dV ). Furthermore, a weighted Lichnerowicz formula holds, which was observed by Perelman [P,Rem. 1.3]. To state it, let be the weighted Laplacian acting on spinors and let be Perelman's weighted scalar curvature (or P-scalar curvature).
Proof. The proof is a consequence of the standard Lichnerowicz formula and the properties of Clifford multiplication. Recall that if e 1 , . . . , e n is a local orthonormal basis of T M , then for any symmetric 2-tensor A, (The proof is immediate from the Clifford algebra relation e i · e j + e j · e i = −2δ ij ½). Combined with the standard Lichnerowicz formula and the Clifford algebra relation, it follows that for any smooth spinor ψ, Remark 1.12. The weighted Lichnerowicz formula also follows from the Lichnerowicz formula for spin-c Dirac operators [Fr2,§3.3], by choosing the spin-c connection ∇ A for which A = − 1 2 df . Indeed, with this connection, (1.14) and dA = − 1 2 d 2 f = 0, from which the weighted Lichnerowicz formula (1.9) follows immediately. In this sense, the weighted Dirac operator can also be thought of as the twisted Dirac operator D A . Proposition 1.15 (Weighted Ricci identity). The weighted Ricci curvature Ric f = Ric + Hess f is proportional to the commutator of D f and ∇: for any vector field X and spinor ψ, Proof. Recall the unweighted Ricci identity, [D, ∇ X ] = 1 2 Ric(X)·. (For a proof, see for example [BHM+,Rem. 2.50]). Using this identity and the fact that Clifford multiplication is parallel with respect to the weighted spin connection (1.4), it follows that, for any spinor ψ, In what follows, denote the space of weighted L 2 -spinors by L 2 f = L 2 (ΣM, e −f dV ) and let L 2 be the space of unweighted L 2 -spinors. Define the linear operator This operator is an isomorphism of Hilbert spaces with inverse given by In Proof. For any C 1 -spinor ψ, This proves (1.21), and it follows immediately from this equation and the fact that U f is an isomorphism, that Dψ = λψ if and only if D f (U f ψ) = λU f ψ. In particular, U f is an isomorphism between the eigenspaces E λ (D) and E λ (D f ), for any λ ∈ R. Hence, (when defined) the multiplicities of the eigenvalues coincide.
The following eigenvalue inequality is a generalization of Friedrich's inequality [Fr1] and the proof below generalizes his proof. See [Fr2,§5.1] for an insightful exposition of the classical proof, whose outline will be followed below. The weighted Friedrich inequality proved below is sharp. Indeed, on the round sphere with constant scalar curvature R and with f a constant function, equality is obtained.
A calculation employing a local orthonormal frame shows that the assumption Dψ = λψ implies Integrating the above equation over M and integrating the last term by parts implies The standard (unweighted) Lichnerowicz formula, the self-adjointness of D on L 2 , and the definition of the weighted scalar curvature R f then implŷ which, after rearranging, implies This was to be shown. If equality occurs in the previous inequality, then R f is constant, ∇ f,λ ψ = 0 and ∇f = 0. In particular, f is constant, so 0 = ∇ f,λ ψ = ∇ 0,λ ψ. This is equivalent to the condition that for all vector fields X. Hence ψ is a Killing spinor. Finally, a manifold admitting a Killing spinor must be Einstein; see for example [Fr2,§5.2]. The converse is immediate.
Whenever the scalar curvature is not constant, Theorem 1.23 implies a strict improvement of Friedrich's inequality. This is because the weight f can always be chosen to make R f constant, while if R is not constant, then it follows that R f > R min . To show this, recall that Perelman's entropy λ P is defined as the first eigenvalue of the operator −4∆ + R, or equivalently, as the minimum of the weighted Hilbert-Einstein functional [P]: If f is the minimizer of λ P , the weighted scalar curvature is constant, with R f = λ P . On the other hand, if the scalar curvature is not constant, then R f = λ P > R min , and thus the weighted Friedrich inequality (1.24) implies a strict improvement of Friedrich's inequality.
Corollary 1.32. Any eigenvalue λ of the Dirac operator D on a closed manifold (M n , g) satisfies with equality if and only if (M n , g) admits a Killing spinor, in which case (M n , g) is Einstein.
The bound (1.32) gives another proof of the stability of hyperkähler metrics on the K3 surface along Ricci flow. Indeed, all metrics on K3 satisfy the above inequality with λ = 0 sinceÂ(K3) = 0. Consequently, Corollary 1.32 implies that λ P (g) 0 for all metrics g on K3, with equality exactly on hyperkähler metrics. These metrics are consequently stable by [Ha1].
In this sense, Hijazi's inequality [Hi,Eqn. (5.1)] is sharper than the inequality of Theorem 1.23. On the other hand, the inequality in Corollary 1.32 improves along Ricci flow.

Weighted asymptotically Euclidean manifolds
A smooth orientable Riemannian manifold (M n , g) is called asymptotically Euclidean (AE) of order τ if there exists a compact subset K ⊂ M and a diffeomorphism Φ : M \ K → R n \ B ρ (0), for some ρ > 0, with respect to which (2.1) for any partial derivative of order k as r → ∞, where r = |Φ| is the Euclidean distance function. The set M \ K is called the end of M n . (The results of this section extend in a straightforward manner to AE manifolds with multiple ends, though they are not pursued here.) The ADM mass [ADM] of (M n , g) is defined by is a coordinate sphere of radius ρ. 1 Although the definition of mass involves a choice of AE coordinates, if τ > (n − 2)/2 and the scalar curvature is integrable, then the mass is finite and independent of the choice of AE coordinates [B, C]. If n ≤ 7 or (M n , g) admits a spin structure, then the assumptions R ≥ 0, R ∈ L 1 (M, g), and τ > n−2 2 , imply that m(g) is nonnegative and is zero if and only if (M n , g) is isometric to (R n , g euc ), by the positive mass theorem [SY,W1].
The AE structure defines a trivialization of the spin bundle at infinity. Indeed, choose an asymptotic coordinate system Φ −1 : R n \ B R (0) → M \ K. The pullback bundle (Φ −1 ) * ΣM differs from the trivial spin bundle R n × Σ by an element of H 1 (R n \ B R (0); Z) = 0. Hence the spin structure is trivial over the end of M and the bundle (Φ −1 ) * ΣM extends trivially over all of R n . This trivialization of the spin bundle allows for the definition of "constant spinors" on the end of M : a spinor ψ defined on the end M is called constant (with respect to the asymptotic coordinates Φ) if ψ = (Φ −1 ) * ψ 0 , for some constant spinor ψ 0 on R n .
Witten argued that for any such constant spinor ψ 0 on M \ K with |ψ 0 |→ 1 at infinity, there exists a harmonic spinor ψ on M which is asymptotic to ψ 0 , in the sense that |ψ − ψ 0 |= O(r −τ ) and |∇ψ|= O(r −τ −1 ). Such a spinor ψ is called a Witten spinor. Moreover, the ADM mass of (M n , g) is given by which is called Witten's formula for the mass. A rigorous proof of the existence of Witten spinors is given by Parker-Taubes [PT] and Lee-Parker [LP]; their proofs are generalized below and in Appendix A.
2.1. Weighted mass. The weighted ADM mass of a weighted AE manifold (M n , g, f ) is defined by This definition is motivated by the weighted Witten formula (2.7) below, and manifestly extends to non-spin manifolds. Like ADM mass, the weighted mass is independent of the choice of asymptotic coordinates if τ > n−2 2 and R ∈ L 1 (M ): indeed, the ADM mass is coordinate independent under said assumptions [B, C], and by the divergence theorem, the second term in (2.4) equals 2´M (∆ f f ) e −f dV , which is manifestly coordinate independent.
The appropriate analytic tools for studying AE manifolds are the weighted Hölder spaces C k,α β (M ), whose precise definitions are stated in Appendix A. These spaces share many of the global elliptic regularity results which hold for the usual Hölder spaces on compact manifolds. The index β is important because it denotes the order of growth: functions in C k,α β (M ) grow at most like r β . In particular, if the metric g is AE of order τ on M = R n , then in the AE coordinate system, g − δ lies in C k,α −τ (M ) for all k ∈ N and the scalar curvature of g lies in C k,α −τ −2 (M ) for all k ∈ N. In what follows, let D f be the weighted Dirac operator associated with the weighted spin connection (1.2) defined by f , which satisfies the weighted Lichnerowicz formula (1.9).
it follows that By the definition of the weighted scalar curvature (1.7) and Witten's formula for the mass, Integrating the last term by parts and using the fact that |ψ f |→ 1 at infinity gives By the assumption f → 0 at infinity and |∇f |= O(r −(n−1) ), the latter limit exists and is finite, since the area of S ρ is of order ρ n−1 .
The following theorem generalizes Schoen-Yau [SY] and Witten's [W1] positive mass theorem to the weighted (spin) setting.
Theorem 2.13 (Positive weighted mass theorem). Let (M n , g, f ) be a weighted, spin, AE manifold satisfying the assumptions of Theorem 2.5. Then m f (g) 0, with equality if and only if (M n , g) is isometric to (R n , g euc ) and´R n (∆ f f ) e −f dV = 0.
Proof. Theorem 2.5 provides the existence of a weighted Witten spinor ψ satisfying the weighted Witten formula (2.7). This shows that m f (g) ≥ 0 if R f ≥ 0. The proof of the equality statement resembles Witten's proof of the equality statement for the positive mass theorem: equality implies that ∇ψ = 0, and since there exist rank(ΣM ) possible linearly independent constant spinors at infinity ψ 0 to which ψ is asymptotic, ΣM admits a basis of parallel spinors. Since the map ΣM → T M sending a spinor ϕ to the vector field V ϕ defined by (2.14) V ϕ , X = Im ϕ, X · ϕ for all X ∈ Γ(T M ), is surjective, and since V ϕ is a parallel vector field if ϕ is a parallel spinor, T M admits a basis of parallel vector fields. Thus (M n , g) is flat. Finally, since m(g euc ) = 0, integration by parts and m f (g euc ) = 0 imply that 2.2. Weighted mass and Ricci flow. Given an asymptotically Euclidean manifold (M n , g), define the renormalized Perelman entropy as Note that λ ALE (g) can equivalently be defined as the infimum of´M R f e −f dV − m f (g), over all f ∈ C ∞ c (M ). If (M n , g) admits a Witten spinor ψ, then testing the right-hand-side of the above equation with u = |ψ| gives that λ ALE (g) ≤ 0, by Kato's inequality, |∇|ψ||≤ |∇ψ|. As mentioned in the Introduction, Ricci flow is the gradient flow of λ ALE on AE manifolds and λ ALE has various advantages over the ADM mass in the context of Ricci flow; see the Introduction and also [DO1].
Theorem 2.17. Let (M n , g) be an asymptotically Euclidean manifold of order τ > n−2 2 and with nonnegative scalar curvature. Then there exists a solution f ∈ C 2,α −τ (M ) of the elliptic equation R f = 0, and the f -weighted mass satisfies Proof. By [DO1,(2.3)], there exists a strictly positive minimizer w = e −f /2 of (2.16) with w − 1 ∈ C 2,α −τ (M ) satisfying −4∆w + R w = 0. Since w → 1 at infinity, integration by parts implies The result now follows immediately from the definition (2.4) of m f (g) and that of λ ALE , (2.16).
Note that [DO1,Eqn. (2.3)] is stated for ALE manifolds in the neighborhood of a Ricci-flat ALE manifold, to ensure the existence and uniqueness of f by the positivity of −4∆+R thanks to a Hardy inequality; see [DO1,Prop. 1.12]. However, the same proof holds under the above assumptions on (M n , g) since the scalar curvature is nonnegative and the operator −4∆ + R is therefore positive; see the proof of [Ha1,Thm. 2.6] for a similar argument.
It has been proven in [Li,Thm. 2.2] that the AE conditions are preserved along Ricci flow (with the same coordinate system) as long as the flow is nonsingular. An asymptotically Euclidean Ricci flow is defined to be any Ricci flow starting at an AE manifold. In particular, m f (g) is monotone decreasing along the Ricci flow, and is constant only if (M n , g(t)) is Ricci-flat.
Proof. Since m f (g) = −λ ALE (g), equation (2.21) follows from the formula for the first variation of λ ALE , which can be found in [DO1,Prop. 2.3 and 3.13]. Once again, the assumptions of closeness to a Ricci-flat ALE metric of Deruelle-Ozuch can be replaced by the nonnegativity of scalar curvature. Their closeness assumption is again only used to ensure the existence of f . Note that in contrast with Perelman's monotonicity for closed manifolds, which is proved by letting f evolve parabolically backwards in time, the monotonicity formula (2.21) uses the fact that f solves the elliptic equation R f = 0 at each time.
To prove the equality statement, note that formula (2.21) implies that m f (g) is constant if and only if (M n , g, f ) is a steady Ricci soliton. The proof is completed by using [DK,Prop. 2.6]: any ALE steady soliton with ∇f → 0 at infinity is Ricci flat.
As explained in Section 2, the asymptotically Euclidean structure defines a trivialization of the spin bundle at infinity, allowing for the notion of a spinor which is "constant" in the asymptotic coordinate system. In what follows, for ρ > 0, let S ρ = r −1 (ρ) be the ρ-level set of r, that is, a coordinate sphere of radius ρ.
Proof of Theorem 2.5. With respect to the trivialization of the spin bundle at infinity, the weighted Dirac operator may be written as Choose a spinor ψ 0 which is constant at infinity and with |ψ 0 |→ 1 at infinity, and extend it to a smooth spinor on M . It follows from the above equation and the assumption f , so integrating by parts as in the proof of the Lemma above shows that ϕ = 0. Thus ψ is a weighted harmonic spinor which is asymptotic to ψ 0 .
Let X be the vector field on M \ K defined by (A.7) X = Re ∇ i ψ, ψ e −f e i .
Let λ i = Re ∇ i ψ, ψ e −f so that X = λ i e i . Define the (n − 1)-form Then dα = div g (X)dV g and div g (X) = λ i div g (e i ) + ∇λ i , e i (A.9)
Finally, a coordinate calculation shows that