Some triviality results for quasi-Einstein manifolds and Einstein warped products
aa r X i v : . [ m a t h . DG ] F e b SOME TRIVIALITY RESULTS FOR QUASI-EINSTEINMANIFOLDS AND EINSTEIN WARPED PRODUCTS
PAOLO MASTROLIA AND MICHELE RIMOLDI
Abstract.
In this paper we prove a number of triviality results forEinstein warped products and quasi-Einstein manifolds using differenttechniques and under assumptions of various nature. In particular weobtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems. Introduction A weighted manifold , also known in the literature as a smooth metricmeasure space , is a triple ( M m , g M , e − f d vol), where M m is a complete m -dimensional Riemannian manifold with metric g M , f ∈ C ∞ ( M ) and d voldenotes the canonical Riemannian volume form on M . The Ricci tensorcan be naturally extended to weighted manifolds introducing the modified k -Bakry-Emery Ricci tensor (1) Ric kf = Ric + Hess( f ) − k df ⊗ df, for 0 < k ≤ ∞ . When f is constant, Ric kf ≡ Ric , while, if k = ∞ , Ric kf = Ric f , the usualBakry-Emery Ricci tensor. For a detailed introduction to weighted manifoldsand the k -Bakry-Emery Ricci tensor, we refer to the papers of Wei and Wylie([28], [29]) and Li ([12]).We call a weighted manifold k -quasi-Einstein or simply quasi-Einstein (and g M is a quasi-Einstein metric ) if(2) Ric kf = λg M , for some λ ∈ R (see [6]). We note that: • if f = constant, (2) is the Einstein equation, and in this case we callthe quasi-Einstein metric trivial ; Mathematics Subject Classification.
Key words and phrases.
Einstein warped products, quasi-Einstein manifolds, triviality,gradient estimates. • if k = ∞ , (2) is exactly the gradient Ricci soliton equation. In thelast years, since the appearance of the seminal works of R. Hamil-ton [9] and G. Perelman [16], the study of Ricci solitons (and oftheir generalizations) has become the matter of a rapidly increas-ing investigation, directed mainly toward problems of classification and triviality ; among the enormous literature on the subject we onlyquote, as a few examples, the papers [17], [19], [18], [24], [20], [8].In the following we deal only with the case k ∈ N , which corresponds to thecase of Einstein warped product metrics . Indeed, in [6], elaborating on [11],it is proved a characterization of quasi-Einstein metrics as base metrics ofEinstein warped product metrics. This characterization can be formulatedin the following form (see [26], Theorem 2). Recall that the f -Laplacian of aweighted manifold ( M, g M , e − f d vol) is defined as the diffusion-type operator∆ f = e f div( e − f ∇ ). Theorem 1. If N m + k = M m × u F k is a complete Einstein warped productwith Einstein constant λ , warping function u = e − f/k and Einstein fibre F k ,then the weighted manifold ( M m , g M , e − f d vol) satisfies the quasi-Einsteinequation (2) ; furthermore, the Einstein constant µ of the fibre satisfies theequation (3) ∆ f f = kλ − kµe k f . Conversely, if the weighted manifold ( M m , g M , e − f d vol) satisfies (2) , then f satisfies (3) for some constant µ ∈ R . Consider the warped product N m + k = M m × u F k , with u = e − f/k , and Einstein fibre F with Einstein constant µ .Then N is Einstein with Ric N = λg N . The previous characterization permits to study Einstein warped productsby focusing only on equation (3).Examples of quasi-Einstein manifolds with λ < µ of arbitrary sign,or with λ = 0 and µ ≥ µ >
0, while the trivial quasi-Einstein metricswith λ = 0 necessarily satisfy µ = 0. Other non-trivial examples with λ > k > µ > k < ∞ and λ > M isnecessarily compact (see [25]), the maximum principle applied to (3) yieldsthat µ > λ < UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 3 allow us to obtain a triviality result when the function f is bounded frombelow by a constant depending on m , k and on the Einstein constants λ and µ , respectively of the warped product and of the fibre. Further trivialityresults adapting Liouville-type theorems from [15] are also given.In [26] one of the authors prove a triviality result under weighted inte-grability conditions on f . In Section 3, using a Motomiya-type theorem, weare able to obtain the same conclusion under a more natural integrabilityassumption.In Section 4 we concentrate on 1-quasi-Einstein manifolds. Indeed, bymeans of an adaptation to the weighted Laplacian of a Liouville-type resultobtained in [22], we get the triviality of the quasi-Einstein structure or theconstancy of the scalar curvature.2. Gradient estimates and triviality results
A generalization of Case’s gradient estimate.
In [5] J. Case dealswith the triviality of quasi-Einstein metrics, and hence, of Einstein warpedproducts, by considering only equation (3). However in that work only thecase λ ≥ Theorem 2. (Case) Let N m + k = M m × u F k be a complete warped productwith warping function u ( x ) = e − f ( x ) k , scalar curvature N S ≥ and completeEinstein fibre F . Then N is simply a Riemannian product provided the basemanifold M is complete and the scalar curvature of F satisfies F S ≤ . Remark 3. If λ >
0, as observed above, the assertion of Theorem 2 can beeasily proved from (3) using the maximum principle.Note also that in [26] a generalization of Theorem 2 is obtained, proving thetriviality when the warped product and the fibers, respectively, have non-positive and non-negative scalar curvature, up to assume an integrabilitycondition on the warping function u .The proof of Theorem 2 is a consequence of the following gradient estimatefor solutions of weighted Poisson equations (see also [27]).We denote with B ( q, T ) the geodesic ball centered in q of radius T . Theorem 4. (Case) Let ( M m , g M , e − f d vol) be such that Ric kf ≥ , k < ∞ ,and (4) ∆ f f = φ ( f ) , where φ : R → R is a function such that (5) φ ′ ( t ) + 2 m φ ( t ) ≥ PAOLO MASTROLIA AND MICHELE RIMOLDI for all t ∈ R . Then for all q ∈ M , T > such that B ( q, T ) is geodesicallyconnected in M and the closure B ( q, T ) is compact, (6) |∇ f | ( q ) ≤ mkm + k m + k + 6) T . We are able to obtain a similar estimate even in case λ < Theorem 5.
Let ( M m , g M , e − f d vol) be a weighted manifold (not necessarilycomplete). Suppose that, for some k < + ∞ , Z > , (7) Ric kf ≥ λ = − ( m + k − Z and that (8) ∆ f f = ψ ( f ) , where ψ : R → R satisfy (9) ψ ′ ( t ) + 2 m ψ ( t ) + λ ≥ for all t ∈ R . Then for all q ∈ M and T > such that B ( q, T ) is geodesicallyconnected in M and the closure B ( q, T ) is compact, (10) |∇ f | ( q ) ≤ mkm + k " m + k + 6) T − √ λZ T . Remark 6.
Note that in case
Ric kf ≥ we recover Case’s result by letting Z → + .Proof. From the Bochner formula (see, for instance, [15])12 ∆ f |∇ u | = | Hess ( u ) | + g M ( ∇ ∆ f u, ∇ u ) + Ric kf ( ∇ u, ∇ u ) + 1 k g M ( ∇ f, ∇ u ) . Applying the previous formula to f and using (7), (8), (9), Newton inequal-ities and ∆ f = ∆ f f + |∇ f | we obtain12 ∆ f |∇ f | = | Hess ( f ) | + g M ( ∇ ∆ f f, ∇ f ) + Ric kf ( ∇ f, ∇ f ) + 1 k |∇ f | ≥ | Hess ( f ) | + ψ ′ ( f ) |∇ f | + λ |∇ f | + 1 k |∇ f | ≥ m (∆ f ) + ψ ′ ( f ) |∇ f | + λ |∇ f | + 1 k |∇ f | = 1 m ψ ( f ) + (cid:18) ψ ′ ( f ) + 2 m ψ ( f ) + λ (cid:19) |∇ f | + (cid:18) m + 1 k (cid:19) |∇ f | ≥ (cid:18) m + 1 k (cid:19) |∇ f | , UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 5 and then we deduce(11) ∆ f |∇ f | ≥ (cid:18) m + 1 k (cid:19) |∇ f | . Let now ρ ( x ) := dist ( q, x ) (using the Calabi trick, [3], we can suppose that ρ is smooth) and consider on B ( q, T ) the function(12) F ( x ) = (cid:2) T − ρ ( x ) (cid:3) |∇ f | . If |∇ f | ≡ |∇ f | 6≡
0, since F ≥ F | ∂B ( q,T ) ≡
0, there exists a point x ∈ B ( q, T ) such that F ( x ) = max B ( q,T ) F ( x ) >
0. At x we then have(13) ∇ FF ( x ) = 0 , (14) ∆ f FF ( x ) ≤ . A long but straightforward calculation shows that (13) is equivalent to(15) ∇|∇ f | |∇ f | = 2 ∇ ρ T − ρ at x , while, using (15) and the Gauss lemma, condition (14) is equivalent to(16) 0 ≥ − f ρ T − ρ + ∆ f |∇ f | |∇ f | − ρ ( T − ρ ) at x . From the f -Laplacian comparison theorem (see [25], [14]) we have(17) ∆ f ρ ≤ m + k ) + ( m + k − Zρ ] ;combining (11), (16) and (17) we find, at x ,0 ≥ − m + k ) + ( m + k − Zρ ] T − ρ + 2 (cid:18) m + 1 k (cid:19) |∇ f | − ρ ( T − ρ ) , which implies, multiplying through by ( T − ρ ) , that at x we have(18) 0 ≥ − m + k ) + ( m + k − Zρ ] ( T − ρ ) + 2 (cid:18) m + 1 k (cid:19) F − ρ . The previous relation can be rewritten as(19) 0 ≥ − m + k )( T − ρ ) + 2 (cid:18) m + 1 k (cid:19) F − ρ + H ( ρ ) , PAOLO MASTROLIA AND MICHELE RIMOLDI where H : [0 , T ] → R is defined by H ( ρ ) = 4( m + k − Z ( ρ − T ρ ). Since H assumes its minimum value − √ ( m + k − ZT = √ λ Z T for ¯ t = T √ ,equation (19) implies0 ≥ − m + k ) T + 2 (cid:18) m + 1 k (cid:19) (cid:2) T − ρ ( x ) (cid:3) |∇ f | + 8 √ λ Z T − ρ , and so2 (cid:18) m + 1 k (cid:19) (cid:2) T − ρ ( x ) (cid:3) |∇ f | ≤ m + k + 6) T − √ λ Z T , which easily implies the thesis taking the sup on B ( q, T ). (cid:3) As a corollary we immediately get to the following Liouville-type theoremfor Einstein warped products.
Theorem 7.
Let N = M m × u F k a complete Einstein warped product withwarping function u = e − f/k . scalar curvature N S = ( m + k ) λ < andcomplete Einstein fibre F k with scalar curvature F S = kµ < . Suppose that (20) f ≥ k (cid:18) λ µ m + 2 km + k (cid:19) for all x ∈ M. Then N is simply a Riemannian product (up to a rescaling of the metric on F ).Proof. Since N is an Einstein warped product, from Theorem 1 we knowthat f satisfies the equation∆ f f = kλ − kµe k f , so, with the notation used above, we have that ψ ( t ) = kλ − kµe k t . Equation(20) implies (9), so we can apply Theorem 5. Since M is complete, letting T → + ∞ we obtain the thesis. (cid:3) Applications of other gradient estimates.
In the same spirit, adapt-ing results from [15] we also achieve other triviality results from a-priori es-timates for the gradient of global solutions of equations slightly more generalthan (4). In particular as a consequence of Theorem 2.3 in [15] we deducethe following
Theorem 8.
Let N = M m × u F k a complete Einstein warped product withEinstein constant λ < , warping function u = e − f/k and Einstein fibre F k with Einstein constant µ < . Suppose that (21) f ≥ k (cid:18) λ µ (cid:19) for all x ∈ M UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 7 and that (22) | f | ≤ D (1 + r ( x )) ν for some D ≥ , ν ∈ R . Then N is simply a Riemannian product(up to arescaling of the metric on F ), provided (23) 0 ≤ ν < . Proof.
Since N is an Einstein warped product, from the previous discussionswe know that f satisfies (3). Now, referring to Theorem 2.3 in [15], condition(2.23) is satisfied (with equality sign) for δ = 0 and λ = − ( n − H = − ( m + k − H , condition (2.25) is guaranteed by (21) and (2.26) is validfor all θ ∈ R , since A = B = e − f , so we can choose, for instance, θ = − f is constant by Theorem 2.3 in [15]. (cid:3) Remark 9.
Note that condition (21) is more general than (20) , but in orderto obtain triviality in Theorem 8 we also need to require f to have sublineargrowth. A refined version of Theorem 1 in [26]In the present section we state a weighted version of Theorem 1.31 in [23],which can be proved by minor changes to the proof of this latter, and a suffi-cient condition for the validity of the full Omori-Yau maximum principle forthe f -Laplacian; our goal is to deduce a triviality result for complete Einsteinwarped products, which is a corollary of Theorem 1 in [26], replacing the in-tegrability assumption with weight e − fk in the aforementioned theorem witha more natural condition. We recall that a Riemannian manifold ( M, h , i )is said to satisfy the Omori-Yau maximum principle for the f -laplacian iffor each u ∈ C ( M ) such that u ∗ = sup M u < + ∞ there exists a sequence { x k } ⊂ M such that( i ) u ( x k ) > u ∗ − k , ( ii ) |∇ u ( x k ) | < k , ( iii ) ∆ f u ( x k ) < k for each k ∈ N . Theorem 10.
Assume on the complete weighted manifold ( M, g M , e − f d vol) the validity of the full Omori-Yau maximum principle for the f -Laplacian.Let v ∈ C ( M ) be a solution of the differential inequality ∆ f v ≥ Φ( v, |∇ v | ) , with Φ( t, y ) continuous in t , C in y and such that ∂ Φ ∂y ( t, y ) ≥ . PAOLO MASTROLIA AND MICHELE RIMOLDI
Set ϕ ( t ) = Φ( t, . Then a sufficient condition to guarantee that v ∗ = sup M v < + ∞ is the existence of a continuous function F positive on [ a, + ∞ ) for some a ∈ R satisfying (24) (cid:26)Z ta F ( s ) ds (cid:27) − ∈ L (+ ∞ ) , (25) lim sup t → + ∞ R ta F ( s ) dstF ( t ) < + ∞ , (26) lim inf t → + ∞ ϕ ( t ) F ( t ) > and (27) lim inf t → + ∞ nR ta F ( s ) ds o F ( t ) ∂ Φ ∂y (cid:12)(cid:12)(cid:12)(cid:12) ( t, > −∞ . Furthermore in this case ϕ ( v ∗ ) ≤ . Consider now the equation(28) ∆ f f = kλ − kµe k f . and let µ <
0. If we choose ϕ ( t ) = Φ( t, y ) = mλ − mµe k t and F ( t ) = ( t − a ) σ ,with σ >
1, then F satisfies the assumptions of Theorem 10. However, touse Theorem 10, we have also to assure on ( M, g M , e − f d vol) the validity ofthe full Omori-Yau maximum principle for the f -laplacian.We will use the following corollary of Theorem 4.1 in [20]. Corollary 11.
Let ( M m , g M , e − f d vol) be a complete weighted manifold suchthat (29) Ric kf ( ∇ r, ∇ r ) ≥ − ( m + k − G ( r ) for a smooth positive function G on [0 , + ∞ ) , even at the origin and satisfying (30) ( i ) G (0) > ii ) G ′ ( t ) ≥ on [0 , + ∞ )( iii ) G ( t ) − / ∈ L (+ ∞ ) ( iv ) lim sup t → + ∞ tG (cid:16) t (cid:17) G ( t ) < + ∞ . Then the Omori-Yau maximum principle for the f -Laplacian holds on M . UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 9
Proof.
Let h be the solution on R +0 of the Cauchy problem (cid:26) h ′′ − Gh = 0 h (0) = 0; h ′ (0) = 1 . Then, by Proposition 2.3 in [14], the inequality∆ f r ≤ − ( m + k − h ′ h ≤ C G ( r ) , holds pointwise in M \ ( cut ( o ) ∪ { o } ) for some constant C . Thus(31) ∆ f r = 2 r ∆ f r + 2 ≤ rC G ( r ) ≤ C rG ( r ) , off a compact set, and the hypotheses (4.1), (4.2) and (4.3) of Theorem 4.1in [20] are satisfied with γ = r . In that theorem it is also assumed a boundon the gradient of f , but here we don’t need this further hypothesis. Indeedby (31) we can replace the last part of the proof of Theorem 4.1 in [20] withthe following computation.∆ f u ( x k ) = ∆ u ( x k ) − h∇ u, ∇ f i ( x k ) ≤ ( u ( x k ) − u ( p ) + 1) k ( ϕ ′ ( γ ( x k )) ϕ ( γ ( x k )) ∆ γ ( x k ) + 1 k (cid:18) ϕ ′ ( γ ( x k )) ϕ ( γ ( x k )) (cid:19) |∇ γ | ( x k ) ) − ( u ( x k ) − u ( p ) + 1) k ϕ ′ ( γ ( x k )) ϕ ( γ ( x k )) h∇ γ ( x k ) , ∇ f ( x k ) i≤ ( u ( x k ) − u ( p ) + 1) k ( ϕ ′ ( γ ( x k )) ϕ ( γ ( x k )) ∆ f γ ( x k ) + 1 k (cid:18) ϕ ′ ( γ ( x k )) ϕ ( γ ( x k )) (cid:19) |∇ γ | ( x k ) ) ≤ ( u ( x k ) − u ( p ) + 1) k ( cγ / G (cid:0) γ / (cid:1) / C γ / G (cid:16) γ / (cid:17) / + 1 k · c γG (cid:0) γ / (cid:1) A γ ) , and the RHS tends to zero as k → + ∞ . (cid:3) Hence, choosing G ( t ) = t + | λ | + εm + k − , for some ε >
0, we obtain that thefull Omori–Yau maximum principle for the f –laplacian holds on a genericquasi–Einstein manifold.As an application of Theorem 10 we can deduce the following result. Corollary 12.
Let N m + k = M m × u F k be a complete Einstein warped prod-uct with non-positive scalar curvature ( m + k ) λ = N S ≤ , warping function u ( x ) = e − f ( x ) k satisfying inf M f = f ∗ > −∞ and complete Einstein fibre F . Suppose also that F S < . Then f ∗ < + ∞ . In particular Riemannianvolumes are equivalent to f –weighted volumes. Proof.
Applying Theorem 10 to equation (28) we obtain that f ∗ < + ∞ .Since, by assumption, we know also that f ∗ > −∞ the thesis follows easily. (cid:3) From Corollary 12 we immediately get the following corollary of Theorem1 in [26].
Corollary 13.
Let N m + k = M m × u F k be a complete Einstein warpedproduct with non-positive scalar curvature ( m + k ) λ = N S ≤ , warpingfunction u ( x ) = e − f ( x ) k satisfying inf M f = f ∗ > −∞ and complete Einsteinfibre F . Suppose also that F S < . Then N is simply a Riemannian productif the base manifold M is complete and non-compact, the warping functionsatisfies f ∈ L p ( M, e − f d vol) , for some < p < + ∞ , and f ( x ) ≤ forsome point x ∈ M . From the Motomiya–type theorem we deduce also the following result.
Theorem 14.
Let N m + k = M m × u F k be a complete Einstein warped prod-uct with non–positive scalar curvature ( m + k ) λ = N S ≤ , warping function u ( x ) = e − f ( x ) k satisfying inf M f = f ∗ > −∞ and complete Einstein fibre F with F S < . Then M S ∗ = mλ .Proof. As above, by Theorem 10, we have that f ∗ < + ∞ and so vol b f ( M ) ≤ vol f ( M ) e k − k f ∗ From the weighted volume estimates in [25] and Theorem 9 in [24] we getthat the weak maximum principle at infinity for the b f -Laplacian holds on M . Hence we can construct a sequence { x n } such that f ( x n ) → f ∗ and∆ b f f ( x n ) ≥ − n . Thus, since tracing (2) we have that ∆ b f f = mλ − M S , weobtain that − n ≤ mλ − M S ( x n ) ≤ mλ − M S ∗ ≤ , where in the last inequality we have used the estimates of Theorem 3 in [26].The conclusion now follows taking the limit for n → + ∞ . (cid:3) A Liouville result for 1-quasi-Einstein manifolds
In this section we obtain another Liouville result for k -quasi-Einstein man-ifolds, in case k = 1. It is well known that any 1-quasi-Einstein metric whichhas µ = 0 (and so corresponds to a warped product Einstein metric) nec-essarily has constant scalar curvature R ≡ ( m − λ . This follows simplyby taking the trace of the quasi-Einstein equation (2) and using equation(3). Warped product Einstein metrics which correspond to these latters aremore commonly known as static metrics and have been studied extensively UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 11 due to their connections to scalar curvature, the positive mass theorem, andgeneral relativity, (see e.g. [1], [7] and references indicated in the recentpreprint [10]).As observed in [4], also the study of quasi-Einstein metrics with k = 1and µ = 0 is interesting. Since we cannot apply Theorem 1 to construct therelated Einstein warped products, their existence proves that, even restrict-ing to integer hidden dimension k , quasi–Einstein manifolds form a strictlylarger class of manifolds that those which are the base of an Einstein warpedproduct manifold. For some examples of these manifolds, constructed in themore general setting of conformally warped manifolds, see the last sectionof [4].Our Liouville result, which is relevant exactly in the µ = 0 case, willfollow from an adaptation to the f -Laplacian under weighted volume growthconditions of Theorem A in [22]. This can be deduced from the proof of thelatter, making minor modifications in the proofs of Theorem A, Lemma 1.2,Theorem A ′ in [21] and Theorem 2.5 in [22]. Theorem 15.
Let φ be a continuous function on [0 , + ∞ ) satisfying theconditions (32)( i ) φ (0) = φ ( a ) = 0 , ( ii ) φ ( s ) > in (0 , a ) , ( iii ) φ ( s ) < in ( a, + ∞ ) , for some a > , and (33) lim inf s → + ∞ − φ ( s ) s σ > , for some σ > ; let also b ( x ) ∈ C ( M ) and suppose that b ( x ) ≥ C (1 + r ( x )) µ on M, for some C > and ≤ µ < . Let u be a non-negative solution of (34) ∆ f u = − b ( x ) φ ( u ) on M. Assume that (35) lim inf r → + ∞ log vol f ( B r ) r − µ < + ∞ and, if ( vol f ( ∂B r )) − ∈ L (+ ∞ ) assume furthermore that φ ( t ) ≥ ct ξ < t ≪ for some ξ > and c > . Finally, if ξ ≥ suppose also that u ( x ) ≥ Dr ( x ) − θ , r ( x ) ≫ for some θ ≥ , D > and that lim inf r → + ∞ log vol f ( B r ) r − θ ( ξ − ε ) − µ < + ∞ for some ε > . Then u is constant and identically equal to or a . Now, consider a k -quasi-Einstein manifold ( M m , g M , e − f d vol), k < + ∞ .We recall that, according to Lemma 4 in [26], setting e f = k +2 k f , the scalarcurvature S of a quasi-Einstein manifold satisfies the following relation,(36)12 ∆ e f S = − k − k (cid:12)(cid:12)(cid:12)(cid:12) Ric − m Sg M (cid:12)(cid:12)(cid:12)(cid:12) − k + m − km ( S − mλ ) (cid:18) S − m ( m − k + m − λ (cid:19) . Exploiting (36), one can obtain estimates for the infimum of the scalar cur-vature S ∗ = inf M S (see Theorem 3 in [26]). In particular we have that for λ > m ( m − m + k − < S ∗ ≤ mλ, and for λ < M f > −∞ (38) mλ ≤ S ∗ ≤ m ( m − m + k − λ. In the special case k = 1 equation (36) becomes(39) ∆ e f S = − S − mλ )( S − ( m − λ ) . Making an essential use of (39), we now prove the following theorem which,jointly with (37) and (38), essentially states that, under suitable geometricassumption, when the scalar curvature is confined in a particular intervalit has to be constant and identically equal to one of the extremes of theinterval. Some extra rigidity in case λ >
Theorem 16.
Let ( M m , g M , e − f d vol) be a geodesically complete - quasi-Einstein manifold with quasi-Einstein constant λ and scalar curvature S .Set e f = 3 f and suppose that f ∗ = inf M f > −∞ .If ( vol e f ( ∂B r )) − ∈ L (+ ∞ ) , letting u ( x ) = ( − S ( x ) + ( m − λ λ < − S ( x ) + mλ λ > , assume furthermore that u ( x ) ≥ Dr ( x ) − θ , r ( x ) ≫ UASI-EINSTEIN MANIFOLDS AND EINSTEIN WARPED PRODUCTS 13 for some θ ≥ , D > , and that lim inf r → + ∞ log vol e f ( B r ) r − θε < + ∞ for some ε > . (a) If λ < and S ≤ ( m − λ we obtain that S is constant and identicallyequal to either ( m − λ or mλ . (b) If λ > and S ≤ mλ then S is constant, identically equal to mλ and M is Einstein.Proof. (a) Assume λ <
0. Considering u = − S + ( m − λ , which is non-negative for S ≤ ( m − λ , from (39) we obtain that(40) ∆ e f u = 2 u ( u + λ ) . We want now to apply Theorem 15 to the equation (40) on the weightedmanifold (
M, g M , e − e f d vol). If we choose φ ( t ) = − t ( t + λ ), it clearly satisfiesassumptions (32) and (33) with a = − λ and the equation (40) can be writtenin the form ∆ e f u = − φ ( u ) , as in the statement of Theorem 15.Furthermore, according to Qian weighted volume estimates, ([25]), sinceby assumption f ∗ > −∞ , we have the validity of the condition on the e f -volume growth of the form (35). Hence by Theorem 15 we are able toconclude that S is constant and identically equal to either mλ or ( m − λ .(b) Assume λ >
0. Consider u = − S + mλ which is non-negative for S ≤ mλ , and choose φ ( t ) = − t ( t − λ ); applying Theorem 15 with a = λ ,we conclude that S is constant and identically equal to either ( m − λ or mλ .Now we show that the first case cannot happen. Indeed, suppose that S ≡ ( m − λ . Substituing in the trace of the quasi-Einstein equation we getthat ∆ f ≥ M is compact we obtain that f is constant and M isEinstein with Ric = λg M . But this is clearly impossible, since tracing thislatter equation we get a contradiction. Hence S ≡ mλ . Substituing again inthe trace of the quasi-Einstein equation we obtain, reasoning as above, that f is constant, and thus that M is Einstein. (cid:3) Acknowledgements.
We wish to thank Stefano Pigola and Jeffrey Case forvaluable suggestions and useful comments on earlier versions of the paper.
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