Abstract
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.
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1 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 \(f:M\rightarrow \mathbb {R}\), 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 [P], 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.
1.1 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 \(\nabla f\) 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, \(\Delta _f=\Delta -\nabla _{\nabla f}\), also called the drift Laplacian, f-Laplacian, or Witten Laplacian. When \(f=\frac{|x|^2}{4}\) on \(\mathbb {R}^n\), then \(\Delta _f\) 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 \(\lambda \) 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].
1.2 Weighted Asymptotically Euclidean Manifolds
A fundamental quantity associated with an asymptotically Euclidean (AE) manifold \((M^n,g)\) is the ADM mass [ADM], denoted \(\mathfrak {m}(g)\). 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 \(S_{\rho }\) is a coordinate sphere of radius \(\rho \) with outward normal \(\nu \) and area form dA. The normalization for \(\mathfrak {m}\) used in this paper is related to Bartnik’s [B] by \(\mathfrak {m}=c_nm_{\mathrm {ADM}}\), where \(c_n=2(n-1)\omega _{n-1}\) and \(\omega _{n-1}\) is the area of the unit sphere in \(\mathbb {R}^n\); 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 \(\psi \) 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
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 \(\mathrm {SU}(2)\) at infinity, though they are not pursued in this paper.
1.2.1 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 \({\text {R}}_{f_g}\equiv 0\). Theorem 2.17 shows that such an \(f_g\) exists on any AE manifold with nonnegative scalar curvature. This surprisingly yields the formula
where \(\lambda _{\mathrm {ALE}}(g)\) 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 \(\lambda _{\mathrm {ALE}}\) stems from a variational principle on the whole manifold, and a priori is not a boundary term. (The notation for \(\lambda _{\mathrm {ALE}}\) is adopted from [DO1], since the results here also apply to ALE spaces.)
The renormalized Perelman functional is the correct modification of Perelman’s \(\lambda \)-functional (for closed manifolds) to AE manifolds: it has the crucial property that Ricci flow, \(\partial _tg = -2\mathrm {Ric}\), 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):
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 \(\varphi \), the integral \(\int _M(|\nabla \varphi |^2 +\frac{1}{4}{\text {R}}|\varphi |^2)dV\) 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 \(\lambda _{\mathrm {ALE}}\) 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, \(\mathfrak {m}_{f_g}(g)\) is nonnegative on any spin AE manifold, and vanishes only on Euclidean space; \(\mathfrak {m}_{f_g}(g)\) satisfies a Łojasiewicz inequality measuring the distance to Euclidean space; \(\mathfrak {m}_{f_g}(g)\) 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, \(\mathfrak {m}_{f_g}(g)\) is nonnegative and detects how far from Euclidean space the geometry is, allowing for stability analysis of gravitational instantons under general perturbations [DO2].
2 Weighted Dirac Operator
Let \((M^n,g)\) be a complete Riemannian spin n-manifold without boundary. The spin bundle \(\Sigma M\rightarrow M\) is a complex vector bundle of rank \(2^{\lfloor \frac{n}{2}\rfloor }\), 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 \(\Sigma M\). For background on spin geometry, see the book [P], whose notation and conventions are adopted here.
Let \(f\in C^{\infty }(M)\). The weighted Dirac operator \(D_f:\Gamma (\Sigma M)\rightarrow \Gamma (\Sigma M)\) is defined as
where \(D=e_i\cdot \nabla _i\) is the standard (Atiyah-Singer) Dirac operator and \(\cdot \) 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 \(\nabla ^f:\Gamma (\Sigma M)\rightarrow \Gamma (T^*M\otimes \Sigma M)\), defined by
where \(\nabla \) is the standard spin connection induced by the Levi-Civita connection. The modified spin connection \(\nabla ^f\) is not metric compatible with the standard metric [P, Proposition 2.5] on the spin bundle, \(\langle \cdot ,\cdot \rangle \), however, it is compatible with the modified metric \(\langle \cdot ,\cdot \rangle _f := \langle \cdot ,\cdot \rangle e^{-f}\), that is
for any vector field X and spinors \(\psi ,\varphi \). Moreover, since Clifford multiplication is parallel with respect to the standard spin connection, it is also parallel with respect to \(\nabla ^f\). This means that
for any vector fields X, Y and spinor \(\psi \).
The weighted Dirac operator satisfies the following weighted integration by parts formula on \(W^{1,2}(e^{-f}\,dV)\),
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).
Proposition 1.8
(Weighted Lichnerowicz). The square of the weighted Dirac operator \(D_f\) satisfies
Proof
The proof is a consequence of the standard Lichnerowicz formula and the properties of Clifford multiplication. Recall that if \(e_1,\dots ,e_n\) is a local orthonormal basis of TM, then for any symmetric 2-tensor A,
(The proof is immediate from the Clifford algebra relation \(e_i\cdot e_j+e_j\cdot e_i=-2\delta _{ij}\mathbbm {1}\)). Combined with the standard Lichnerowicz formula and the Clifford algebra relation, it follows that for any smooth spinor \(\psi \),
\(\square \)
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 \(\nabla ^A\) for which \(A=-\frac{1}{2}df\). Indeed, with this connection,
and \(dA=-\frac{1}{2}d^2f=0\), 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 \(\mathrm {Ric}_f=\mathrm {Ric}+\mathrm {Hess}_f\) is proportional to the commutator of \(D_f\) and \(\nabla \): for any vector field X and spinor \(\psi \),
Proof
Recall the unweighted Ricci identity, \([D,\nabla _X]=\frac{1}{2}\mathrm {Ric}(X)\cdot \). (For a proof, see for example [Fr1, 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 \(\psi \),
\(\square \)
In what follows, denote the space of weighted \(L^2\)-spinors by \(L_f^2=L^2(\Sigma M,e^{-f}dV)\) 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 \(U_f^{-1}=U_{-f}\); it preserves norms since
In particular, \(U_f\) is a unitary operator. Recall that two operators A, B acting on Hilbert spaces with domains of definition \(\mathcal {D}_A\) and \(\mathcal {D}_B\) are unitarily equivalent if there exists a unitary operator U such that \(U\mathcal {D}_A=\mathcal {D}_B\) and \(UAU^{-1}x=Bx\) for all \(x\in \mathcal {D}_B\).
Proposition 1.20
(Unitary equivalence). The Dirac operator D and the weighted Dirac operator \(D_f\) are unitarily equivalent and hence isospectral; on \(C^1\)-spinors, these operators are related by
In particular, \(D\psi =0\) if and only if \(D_f(e^{f/2}\psi )=0\).
Proof
For any \(C^1\)-spinor \(\psi \),
This proves (1.21), and it follows immediately from this equation and the fact that \(U_f\) is an isomorphism, that \(D\psi =\lambda \psi \) if and only if \(D_f(U_f\psi )=\lambda U_f\psi \). In particular, \(U_f\) is an isomorphism between the eigenspaces \(E_{\lambda }(D)\) and \(E_{\lambda }(D_f)\), for any \(\lambda \in \mathbb {R}\). Hence, (when defined) the multiplicities of the eigenvalues coincide. \(\square \)
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 \({\text {R}}\) and with f a constant function, equality is obtained.
Theorem 1.23
Suppose that \((M^n,g)\) is closed, let \(f\in C^{\infty }(M)\), and let \(\lambda \) be an eigenvalue of the Dirac operator D. Then
with equality if and only if f is constant and \((M^n,g)\) admits a Killing spinor, in which case \((M^n,g)\) is Einstein.
Proof
Let \(\psi \) be an eigenspinor of the Dirac operator with \(D\psi =\lambda \psi \).
Define the connection
A calculation employing a local orthonormal frame shows that the assumption D\(\psi \)=\(\lambda \psi \) 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 then imply
which, after rearranging, implies
This was to be shown.
If equality occurs in the previous inequality, then \({\text {R}}_f\) is constant, \(\nabla ^{f,\lambda }\psi =0\) and \(\nabla f=0\). In particular, f is constant, so \(0=\nabla ^{f,\lambda }\psi =\nabla ^{0,\lambda }\psi \). This is equivalent to the condition that, for all vector fields X
Hence \(\psi \) s a Killing spinor.
Finally, a manifold admitting a Killing spinor must be Einstein; see for example [Fr2, §5.2]. The converse is immediate. \(\square \)
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 \({\text {R}}_f\) constant, while if \({\text {R}}\) is not constant, then it follows that \({\text {R}}_f>{\text {R}}_{\min }\). To show this, recall that Perelman’s entropy \(\lambda _{\mathrm {P}}\) is defined as the first eigenvalue of the operator \(-4\Delta +{\text {R}}\), or equivalently, as the minimum of the weighted Hilbert-Einstein functional [P]:
If f is the minimizer of \(\lambda _{\mathrm {P}}\), the weighted scalar curvature is constant, with \({\text {R}}_f=\lambda _{\mathrm {P}}\). On the other hand, if the scalar curvature is not constant, then \({\text {R}}_f=\lambda _{\mathrm {P}}>{\text {R}}_{\min }\), and thus the weighted Friedrich inequality (1.24) implies a strict improvement of Friedrich’s inequality.
Corollary 1.32
Any eigenvalue \(\lambda \) 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 \(\lambda = 0\) since \(\hat{A}(K3)\ne 0\). Consequently, Corollary 1.32 implies that \(\lambda _P(g)\leqslant 0\) for all metrics g on K3, with equality exactly on hyperkähler metrics. These metrics are consequently stable by [Ha1].
Remark 1.34
Hijazi [Hi, Eqn. (5.1)] proved an inequality closely related to that of Theorem 1.23. Hijazi’s proof employs the Dirac operator of a conformally related metric, whereas the proof of Theorem 1.23 keeps the metric fixed and uses the weighted Lichnerowicz formula (1.9). Hijazi’s inequality implies that any eigenvalue \(\lambda \) of the Dirac operator satisfies \(\lambda ^2\ge \frac{n}{4(n-1)}\mu _1(g)\), where \(\mu _1(g)\) is the smallest eigenvalue of the conformal Laplace operator \(-4\frac{n-1}{n-2}\Delta +{\text {R}}\). Since \(\lambda _{\mathrm {P}}(g)\) is the first eigenvalue of the operator \(-4\Delta +{\text {R}}\), it follows that
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.
3 Weighted Asymptotically Euclidean Manifolds
A smooth orientable Riemannian manifold \((M^n,g)\) is called asymptotically Euclidean (AE) of order \(\tau \) if there exists a compact subset \(K\subset M\) and a diffeomorphism \(\Phi :M\setminus K\rightarrow \mathbb {R}^n\setminus B_{\rho }(0)\), for some \(\rho >0\), with respect to which
for any partial derivative of order k as \(r\rightarrow \infty \), where \(r=|\Phi |\) is the Euclidean distance function. The set \(M\setminus K\) 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
where \(S_{\rho }=r^{-1}(\rho )\) is a coordinate sphere of radius \(\rho \).Footnote 1 Although the definition of mass involves a choice of AE coordinates, if \(\tau >(n-2)/2\) and the scalar curvature is integrable, then the mass is finite and independent of the choice of AE coordinates [B, C]. If \(n\le 7\) or \((M^n,g)\) admits a spin structure, then the assumptions \({\text {R}}\ge 0\), \({\text {R}}\in L^1(M,g)\), and \(\tau >\frac{n-2}{2}\), imply that \(\mathfrak {m}(g)\) is nonnegative and is zero if and only if \((M^n,g)\) is isometric to \((\mathbb {R}^n,g_{\mathrm {euc}})\), 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 \(\Phi ^{-1}:\mathbb {R}^n\setminus B_R(0)\rightarrow M\setminus K\). The pullback bundle \((\Phi ^{-1})^*\Sigma M\) differs from the trivial spin bundle \(\mathbb {R}^n\times \Sigma \) by an element of \(H^1(\mathbb {R}^n\setminus B_R(0);\mathbb {Z})=0\). Hence the spin structure is trivial over the end of M and the bundle \((\Phi ^{-1})^*\Sigma M\) extends trivially over all of \(\mathbb {R}^n\). This trivialization of the spin bundle allows for the definition of “constant spinors” on the end of M: a spinor \(\psi \) defined on the end M is called constant (with respect to the asymptotic coordinates \(\Phi \)) if \(\psi =(\Phi ^{-1})^*\psi _0\), for some constant spinor \(\psi _0\) on \(\mathbb {R}^n\).
Witten argued that for any such constant spinor \(\psi _0\) on \(M\setminus K\) with \(|\psi _0|\rightarrow 1\) at infinity, there exists a harmonic spinor \(\psi \) on M which is asymptotic to \(\psi _0\), in the sense that \(|\psi -\psi _0|=O(r^{-\tau })\) and \(|\nabla \psi |=O(r^{-\tau -1})\). Such a spinor \(\psi \) 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.
3.1 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 \(\tau >\frac{n-2}{2}\) and \({\text {R}}\in L^1(M)\): indeed, the ADM mass is coordinate independent under said assumptions [B, C], and by the divergence theorem, the second term in (2.4) equals \(2\int _M(\Delta _ff) \, e^{-f}dV\), which is manifestly coordinate independent.
The appropriate analytic tools for studying AE manifolds are the weighted Hölder spaces \(C^{k,\alpha }_{\beta }(M)\), 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 \(\beta \) is important because it denotes the order of growth: functions in \(C^{k,\alpha }_{\beta }(M)\) grow at most like \(r^{\beta }\). In particular, if the metric g is AE of order \(\tau \) on \(M=\mathbb {R}^n\), then in the AE coordinate system, \(g-\delta \) lies in \(C^{k,\alpha }_{-\tau }(M)\) for all \(k\in \mathbb {N}\) and the scalar curvature of g lies in \(C^{k,\alpha }_{-\tau -2}(M)\) for all \(k\in \mathbb {N}\).
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).
Theorem 2.5
(Weighted Witten formula). Let \((M^n,g,f)\) be a weighted, spin, AE manifold of order \(\tau \). Suppose that \(f\in C^{2,\alpha }_{-\tau }(M)\), that
and that \(\psi _0\) is a spinor on \((M^n,g)\) which is constant at infinity, with \(|\psi _0|\rightarrow 1\). Then there exists a \(D_f\)-harmonic spinor \(\psi \) which is asymptotic to \(\psi _0\) in the sense that \(\psi -\psi _0\in C^{2,\alpha }_{-\tau }(M)\) and
Proof
Here the proof is given under the additional natural assumptions that \({\text {R}}\ge 0\), \({\text {R}}\in L^1(M,g)\) and \(|\nabla f|=O(r^{-(n-1)})\). The additional assumptions \({\text {R}}\ge 0\), \({\text {R}}\in L^1(M,g)\) ensure the existence of an (unweighted) Witten spinor \(\psi \) [LP]. Further, the assumption \(|\nabla f|=O(r^{-(n-1)})\) is satisfied if \({\text {R}}_f=0\); see [DO1, Prop. (2.2)]. In Appendix A.1, a proof of the general case is given.
By (1.21), if \(D\psi =0\), then the spinor \(\psi _f=e^{f/2}\psi \) is \(D_f\)-harmonic. Since
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 \(|\psi _f|\rightarrow 1\) at infinity gives
By the assumption \(f\rightarrow 0\) at infinity and \(|\nabla f|=O(r^{-(n-1)})\), the latter limit exists and is finite, since the area of \(S_{\rho }\) is of order \(\rho ^{n-1}\). \(\square \)
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 \(\mathfrak {m}_f(g)\geqslant 0\), with equality if and only if \((M^n,g)\) is isometric to \((\mathbb {R}^n,g_{\mathrm {euc}})\) and \(\int _{\mathbb {R}^n}(\Delta _f f)\,e^{-f}dV=0\).
Proof
Theorem 2.5 provides the existence of a weighted Witten spinor \(\psi \) satisfying the weighted Witten formula (2.7). This shows that \(\mathfrak {m}_f(g)\ge 0\) if \({\text {R}}_f\ge 0\). The proof of the equality statement resembles Witten’s proof of the equality statement for the positive mass theorem: equality implies that \(\nabla \psi =0\), and since there exist \(\mathrm {rank}( \Sigma M)\) possible linearly independent constant spinors at infinity \(\psi _0\) to which \(\psi \) is asymptotic, \(\Sigma M\) admits a basis of parallel spinors. Since the map \(\Sigma M\rightarrow TM\) sending a spinor \(\varphi \) to the vector field \(V_{\varphi }\) defined by
is surjective, and since \(V_{\varphi }\) is a parallel vector field if \(\varphi \) is a parallel spinor, TM admits a basis of parallel vector fields. Thus \((M^n,g)\) is flat. Finally, since \(\mathfrak {m}(g_{\mathrm {euc}})=0\), integration by parts and \(\mathfrak {m}_f(g_{\mathrm {euc}})=0\) imply that
\(\square \)
3.2 Weighted Mass and Ricci Flow
Given an asymptotically Euclidean manifold \((M^n,g)\), define the renormalized Perelman entropy as
Note that \(\lambda _{\mathrm {ALE}}(g)\) can equivalently be defined as the infimum of \(\int _M{\text {R}}_f\,e^{-f}dV-\mathfrak {m}_f(g)\), over all \(f\in C^{\infty }_c(M)\). If \((M^n,g)\) admits a Witten spinor \(\psi \), then testing the right-hand-side of the above equation with \(u=|\psi |\) gives that \(\lambda _{\mathrm {ALE}}(g)\le 0\), by Kato’s inequality, \(|\nabla |\psi ||\le |\nabla \psi |\). As mentioned in the Introduction, Ricci flow is the gradient flow of \(\lambda _{\mathrm {ALE}}\) on AE manifolds and \(\lambda _{\mathrm {ALE}}\) 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 \(\tau >\frac{n-2}{2}\) and with nonnegative scalar curvature. Then there exists a solution \(f\in C^{2,\alpha }_{-\tau }(M)\) of the elliptic equation \({\text {R}}_{f}=0\), 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 \(w-1\in C^{2,\alpha }_{-\tau }(M)\) satisfying \(-4\Delta w +{\text {R}}w=0\). Since \(w\rightarrow 1\) at infinity, integration by parts implies
The result now follows immediately from the definition (2.4) of \(\mathfrak {m}_f(g)\) and that of \(\lambda _{\mathrm {ALE}}\), (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 \(-4\Delta +{\text {R}}\) 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 \(-4\Delta +{\text {R}}\) is therefore positive; see the proof of [Ha1, Thm. 2.6] for a similar argument. \(\square \)
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.
Corollary 2.20
(Monotonicity of weighted mass). Let \((M^n,g(t))_{t\in I}\) be an asymptotically Euclidean Ricci flow with nonnegative scalar curvature. Let \(f:M\times I\rightarrow \mathbb {R}\) be the time-dependent family of functions solving \({\text {R}}_{f}=0\) and \(f\rightarrow 0\) at infinity, at each time \(t\in I\). Then
In particular, \(\mathfrak {m}_{f}(g)\) is monotone decreasing along the Ricci flow, and is constant only if \((M^n,g(t))\) is Ricci-flat.
Proof
Since \(\mathfrak {m}_f(g)=-\lambda _{ALE}(g)\), equation (2.21) follows from the formula for the first variation of \(\lambda _{\mathrm {ALE}}\), 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 \({\text {R}}_f=0\) at each time.
To prove the equality statement, note that formula (2.21) implies that \(\mathfrak {m}_f(g)\) 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 \(\nabla f\rightarrow 0\) at infinity is Ricci flat. \(\square \)
Notes
The ADM mass as defined in [B] equals \((2(n-1)\omega _{n-1})^{-1} \mathfrak {m}(g)\), where \(\omega _{n-1}\) is the area of the unit sphere in \(\mathbb {R}^n\).
The meaning of “weighted” in “weighted Hölder spaces” is distinct from its meaning in “weighted manifolds.”
References
Abedin, F., Corvino, J.: On the P-scalar curvature. J. Geom. Anal. 27(2), 1589–1623 (2017)
Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(3), 997 (1961)
Baldauf, J., Ozuch, T. Monotonicity for spinors in the Ricci flow. In preparation
Baldauf, J., Ozuch, T.: Spinors on weighted manifolds with boundary. In preparation
Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)
Bakry, D., Émery, M.: Diffusions hypercontractives. In: Seminaire de Probabilités XIX 1983(84), pp. 177–206. Springer, Berlin, Heidelberg (1985)
Bourguignon, J.P., Hijazi, O., Milhorat, J.L., Moroianu, A., Moroianu, S.: A Spinorial Approach to Riemannian and Conformal Geometry. European Mathematical Society, Berlin (2015)
Branding, V., Habib, G.: Eigenvalue estimates on weighted manifolds. In preparation
Caldarelli, M., Catino, G., Djadli, Z., Magni, A., Mantegazza, C.: On Perelman’s dilaton. Geom. Dedic. 145(1), 127–137 (2010)
Cao, H.D., Zhu, M.: On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353(3), 747–763 (2012)
Chaljub-Simon, A., Choquet-Bruhat, Y.: (1979). Problémes elliptiques du second ordre sur une variété euclidienne à l’infini. In: Annales de la Faculté des sciences de Toulouse: Mathématiques, Vol. 1, No. 1, pp. 9-25
Chruściel, P.: Boundary conditions at spatial infinity. In: Topological properties and global structure of space-time, pp. 49–59. Springer, Boston, MA (1986)
Colding, T.H., Minicozzi, W.P.: (2012). Generic mean curvature flow I; generic singularities. Ann. Math. 755-833
Corvino, J., Pollack, D.: (2011). Scalar curvature and the Einstein constraint equations. arXiv:1102.5050
Dai, X., Ma, L.: Mass under the Ricci flow. Commun. Math. Phys. 274(1), 65–80 (2007)
Deng, J.: Curvature-dimension condition meets Gromov’s n-volumic scalar curvature. SIGMA Symmetry Integr. Geom. Methods Appl. 17, 013 (2021)
Deruelle, A., Kröncke, K.: Stability of ALE Ricci-flat manifolds under Ricci flow. J. Geom. Anal. 31(3), 2829–2870 (2021)
Deruelle, A., Ozuch, T.: (2020). A Łojasiewicz inequality for ALE metrics. arXiv:2007.09937
Deruelle, A., Ozuch, T.: (2021). Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow. arXiv:2104.10630
Fan, E.: Topology of three-manifolds with positive \(P\)-scalar curvature. Proc. Am. Math. Soc. 136(9), 3255–3261 (2008)
Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97(1), 117–146 (1980)
Friedrich, T.: Dirac Operators in Riemannian Geometry, vol. 25. American Mathematical Society, Providence (2000)
Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. de l’IHÉS 58, 83–196 (1983)
Galloway, G.J., Woolgar, E.: Cosmological singularities in Bakry-Émery spacetimes. J. Geom. Phys. 86, 359–369 (2014)
Haslhofer, R.: A renormalized Perelman-functional and a lower bound for the ADM-mass. J. Geom. Phys. 61(11), 2162–2167 (2011)
Haslhofer, R.: A mass-decreasing flow in dimension three. Math. Res. Lett. 19(4), 927–938 (2012)
Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys. 104(1), 151–162 (1986)
Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. (New Series) Am. Math. Soc. 17(1), 37–91 (1987)
Li, C., Mantoulidis, C.: (2021). Metrics with\(\lambda _1 (-\Delta + k )\ge 0\) and flexibility in the Riemannian Penrose Inequality. arXiv:2106.15709
Li, Y.: Ricci flow on asymptotically Euclidean manifolds. Geom. Topol. 22(3), 1837–1891 (2018)
Lichnerowicz, A.: Variétés riemanniennes à tenseur C non négatif. CR Acad. Sci. Paris Sér. AB 271, A650–A653 (1970)
Lichnerowicz, A.: Variétés kählériennes à premiére classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative. J. Differ. Geom. 6(1), 47–94 (1971)
Lott, J., Villani, C.: (2009). Ricci curvature for metric-measure spaces via optimal transport. Ann Math, 903-991
Lu, Y., Minguzzi, E., Ohta, S.I.: Geometry of weighted Lorentz-Finsler manifolds I: singularity theorems. J. Lond. Math. Soc. 104(1), 362–393 (2021)
Munteanu, O., Wang, J.: Analysis of weighted Laplacian and applications to Ricci solitons. Commun. Anal. Geom. 20(1), 55–94 (2012)
Nakajima, H.: (1990). Self-duality of ALE Ricci-flat 4-manifolds and positive mass theorem. In: Recent Topics in Differential and Analytic Geometry. pp. 385-396. Academic Press
Oliynyk, T.A., Woolgar, E.: Rotationally symmetric Ricci flow on asymptotically flat manifolds. Commun. Anal. Geom. 15(3), 535–568 (2007)
Parker, T., Taubes, C.H.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys. 84(2), 223–238 (1982)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)
Schoen, R., Yau, S.T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981)
Sturm, K.T.: On the geometry of metric measure spaces. Acta Math. 196(1), 65–131 (2006)
Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)
Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17(4), 661–692 (1982)
Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)
Woolgar, E., Wylie, W.: Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes. J. Math. Phys. 57(2), 022504 (2016)
Acknowledgements
The first author is indebted to William Minicozzi for continual support, and to Clifford Taubes for inspiring discussions. Part of this work was completed while the first author was funded by a National Science Foundation Graduate Research Fellowship.
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Appendix
Appendix
This appendix provides a proof of the general case of Theorem 2.5 on the existence of a weighted Witten spinor satisfying the weighted Witten formula. In Section 2.1, a simple and illustrative proof was given under natural, albeit more restrictive assumptions.
Let \((M^n,g)\) be an asymptotically Euclidean (AE), Riemannian spin manifold of order \(\tau \). The asymptotic coordinates define a positive function r on M, which equals the Euclidean distance to the origin on \(M\setminus K\), and which can be extended to a smooth function which is bounded below by 1 on all of M.
Using r, the weighted \(C^k\) space \(C^k_{\beta }(M)\) is defined for \(\beta \in \mathbb {R}\) as the set of \(C^k\) functions u on M for which the norm
is finite. The weighted Hölder space \(C^{k,\alpha }_{\beta }(M)\) is defined for \(\alpha \in (0,1)\) as the set of \(u\in C^k_{\beta }(M)\) for which the norm
is finite.Footnote 2 These definitions of weighted Hölder spaces coincide with those of [LP, §9]. In particular, the index \(\beta \) denotes the order of growth: functions in \(C^{k,\alpha }_{\beta }(M)\) grow at most like \(r^{\beta }\). Note that the definitions of the weighted function spaces depend on the “distance function” r, and thereby on the choice of asymptotic coordinates. However, it is easy to see that r is uniformly equivalent to the geodesic distance from an arbitrary fixed point in M as \(r\rightarrow \infty \), hence all choices of r define equivalent norms. For the remainder of this appendix, fix \(\alpha \in (0,1)\).
1.1 Existence of weighted Witten spinors
Let \(f\in C^{\infty }(M)\) and 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).
Lemma A.3
On a weighted, AE, spin manifold \((M^n,g,f)\) satisfying the hypotheses of Theorem 2.5, the operator
is an isomorphism.
Proof
To show injectivity, suppose \(D_f^2\xi =0\) for some \(\xi \in C^{2,\alpha }_{-\tau }(M)\). Then \(\xi =O(r^{-\tau })\) and \(\nabla \xi =O(r^{-\tau -1})\). Applying the weighted Lichnerowicz formula and integration by parts (the boundary term vanishes because \(\tau > (n-2)/2\)), it follows that
Since \({\text {R}}_f\ge 0\), this shows that \(\nabla \xi =0\), so \(\nabla |\xi |^2=0\). Thus \(|\xi |\) is a constant, which is zero since \(\xi \) vanishes at infinity. Thus \(D_f^2\) is injective.
The weighted Lichnerowicz formula implies that \(D_f^2=-\Delta +\nabla _{\nabla f}+\frac{1}{4}{\text {R}}_f\). Since \(f\in C^{2,\alpha }_{-\tau }\) and g is smooth and AE of order \(\tau \), it follows that \(\nabla f\in C^{1,\alpha }_{-\tau -1}\) and \({\text {R}}_f\in C^{0,\alpha }_{-\tau -2}\). Since \(\frac{n-2}{2}<\tau <n-2\), it follows from [CSCB] that \(D_f^2\) is an isomorphism if it is injective. With injectivity proven above, the proof is complete; see [LP, Thm. 9.2d] for the proof for the unweighted Dirac operator. \(\square \)
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 \(\rho >0\), let \(S_{\rho }=r^{-1}(\rho )\) be the \(\rho \)-level set of r, that is, a coordinate sphere of radius \(\rho \).
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 \(\psi _0\) which is constant at infinity and with \(|\psi _0|\rightarrow 1\) at infinity, and extend it to a smooth spinor on M. It follows from the above equation and the assumption \(f\in C^{2,\alpha }_{-\tau }(M)\) that \(D_f^2\psi _0\in C^{0,\alpha }_{-\tau -2}(M)\). By Lemma (A.3), there exists \(\xi \in C^{2,\alpha }_{-\tau }(M)\) with \(D^2_f\xi =D^2_f\psi _0\). The spinor \(\psi =\psi _0-\xi \) then satisfies \(D_f^2\psi =0\) and \(\varphi :=D_f\psi =D_f\psi _0-D_f\xi \) satisfies \(D_f\varphi =0\) and lies in \(C^{1,\alpha }_{-\tau -1}(M)\), so integrating by parts as in the proof of the Lemma above shows that \(\varphi =0\). Thus \(\psi \) is a weighted harmonic spinor which is asymptotic to \(\psi _0\).
Let X be the vector field on \(M\setminus K\) defined by
Let \(\lambda _i=\,\mathrm {Re}\,\langle \nabla _i\psi ,\psi \rangle e^{-f}\) so that \(X=\lambda _ie_i\). Define the \((n-1)\)-form
Then \(d\alpha =\mathrm {div}_g(X) dV_g\) and
hence
Stokes’ theorem then gives, with \(M_{\rho }=\{r\le \rho \}\subset M\) and \(S_{\rho }=\partial M_{\rho }\), that
Since \(\psi =\psi _0-\xi \), the latter boundary term equals
Since \([e_j\cdot ,e_k\cdot ]\) is skew-Hermitian, as in (A.6), it follows that
and so the first term in (A.12) vanishes as \(\rho \rightarrow \infty \). Also, since \(\xi =O(r^{-\tau })\), \(\nabla \xi =O(r^{-\tau -1})\), and \(\nabla \psi _0=O(r^{-\tau -1})\), the third and fourth terms in (A.12) also vanish as \(\rho \rightarrow \infty \). Thus only the second term in (A.12) contributes to the limit \(\rho \rightarrow \infty \); the remainder of the proof consists in showing that said term equals the weighted mass.
To analyze the remaining term, let \(L_i^f\) denote the operator
If \(\beta \) is the \((n-2)\)-form
then since \(e^k\wedge \iota _{e_i}\iota _{e_j} dV_g = \delta _{ik}\iota _{e_j} dV_g-\delta _{jk}\iota _{e_i}dV_g\),
Therefore, by Stokes’ theorem and the fact that \(D_f\xi =D_f\psi _0\), the second term in (A.12) is
Since \(f\rightarrow 0\) at infinity, \(\nabla f= O(r^{\delta -1})\), where \(\delta -1<\tau -(n-1)\) by (2.6), and \(\xi =O(r^{-\tau })\), the last term above vanishes as \(\rho \rightarrow \infty \). Similarly, the second term above vanishes in the limit. On the other hand, (A.6) gives
Writing \(e_i\cdot e_k\cdot = \frac{1}{2}[e_i\cdot , e_k\cdot ] -\delta _{ik}\) and noting that \([e_i\cdot ,e_k\cdot ]\) is skew, it follows that
and hence (A.17) becomes
Putting this into (A.11), letting \(\rho \rightarrow \infty \) and using the definition of mass (2.2) gives the formula
Finally, a coordinate calculation shows that
and since \(|\psi _0|\rightarrow 1\) at infinity, the second-to-last equation gives the weighted Witten formula
\(\square \)
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Baldauf, J., Ozuch, T. Spinors and mass on weighted manifolds. Commun. Math. Phys. 394, 1153–1172 (2022). https://doi.org/10.1007/s00220-022-04420-y
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DOI: https://doi.org/10.1007/s00220-022-04420-y