Rigidity of Quasi-Einstein Metrics

RIGIDITY OF QUASI-EINSTEIN METRICS JEFFREY CASE, YU-JEN SHU, AND G UOF ANG WEI Abstract. W e call a metric quasi-Einstein if the m - Bakr y- Emery Ricci tensor is a c onstan t multiple of the metric tensor. This is a generalization of Einstein metrics, whic h con ta ins gradien t Ricci solitons and is also closely r elated to the construction of the warped pro duct Einstein metrics. W e study pr operties of quasi-Ei nstein metrics and prov e sev eral rigidity results. W e also give a splitting theorem f or some K ¨ ahler quasi-Einstein metrics. 1. Introduction Einstein metr ics and their g eneralizations are imp or tant b oth in mathematics and physics. A particular example is from the study of smo o th metr ic measure spaces. Recall a smooth metric measure space is a triple ( M n , g , e − f dv ol g ), where M is a complete n -dimensiona l Riema nnian manifold with metric g , f is a s mo o th real v alued function on M , a nd dv ol g is the Riemannian volume density on M . A na tural extension o f the Ricci tenso r to smo oth metric mea sure spaces is the m -Bakry -Emery Ricci tensor (1.1) Ric m f = Ric + Hess f − 1 m d f ⊗ d f for 0 < m ≤ ∞ . When f is constant, this is the usua l Ricci tensor . W e call a triple ( M , g , f ) (a Rie- mannian manifold ( M , g ) with a function f on M ) (m-)quas i-Einstein if it satisfies the equation (1.2) Ric m f = Ric + Hess f − 1 m d f ⊗ d f = λg for some λ ∈ R . This e q uation is esp ecia lly interesting in that when m = ∞ it is ex a ctly the gra dient Ricci soliton equa tion; when m is a po sitive integer, it corres p o nds to warp ed pro duct E instein metrics (see Section 2 for detail); when f is constant, it gives the Einstein eq uation. W e call a quasi-Einstein metric trivial when f is consta nt (the rigid case). Many g eometric and top olog ical prop erties of manifolds with Ricci c ur v ature bo unded below can be extended to manifolds with m - Bakry- E mery Ric c i tensor bo unded from b elow when m is finite o r m is infinite a nd f is bo unded, see the survey ar ticle [18] and the references there fo r details. Quasi-E ins tein metrics for finite m and for m = ∞ share some c ommon prop- erties. It is w ell-known now tha t compact solitons with λ ≤ 0 are trivial [8 ]. The same result is prov en in [10] for quasi-E instein metr ic s o n compact manifolds with finite m . Compact shrinking Ricci solitons hav e p ositive sca lar curv a ture [8, 5]. Here we show Pa rtially s upported by NSF grant DMS-0505733. 1 2 JEF FREY CASE, YU-JE N SHU, AND GUOF ANG WE I Prop ositio n 1.1. A quasi-Einstein metric with 1 ≤ m < ∞ and λ > 0 has p ositive sc alar curvatur e. In dimension 2 and 3 compact Ricci so litons are trivial [6, 8]. More generally compact shrinking Ricci solitons with zero W eyl tensor are trivial [5, 15, 13]. W e prov e a similar re s ult in dimension 2 for m finite. Theorem 1. 2. Al l 2-dimensiona l quasi-Einstein metrics on c omp act manifolds ar e trivial. In fact, from the co rresp ondence with warped pro duct metrics, 2-dimensio na l quasi-Einstein metrics (with finite m ) can b e clas sified, s e e [1, The o rem 9 .119]. (The pro of was not published though.) In Section 3 we also extend several prop erties for Ricci s o litons ( m = ∞ ) to quasi-Einstein metrics (genera l m ). On the other hand we show K¨ ahler quasi-Eins tein metrics b ehave very differ ent ly when m is finite a nd m is infinite. Theorem 1.3. L et ( M n , g ) b e an n-dimensional c omplete simply-c onne cte d Rie- mannian manifold with a K¨ ahle r quasi-Einstein metric for finite m . Then M = M 1 × M 2 is a Riemannian pr o duct, and f c an b e c onsider e d as a function of M 2 , wher e M 1 is an (n-2)-dimensional Einstein manifold with Einstein c onst ant λ , and M 2 is a 2-dimensional quasi-Einstein manifold . Combine this with Theorem 1.2 we immediately g et Corollary 1. 4. Ther e ar e no n ontrivial K¨ ahler quasi-Einstein metrics with finite m on c omp act manifolds. Note that all kno wn examples of (non triv ia l) compact s hr inking soliton are K¨ ahler , s ee the survey article [3]. Ricci solito ns play a very imp ortant r ole in the theo ry of Ricci flow and a re exten- sively studied recently . W arp ed pr o duct Einstein metric s hav e co nsiderable interest in physics and many Einstein metrics are constructed in this form, e s pec ially on noncompact manifolds [1]. It was asked in [1] whether o ne could find E instein met- rics with nonco ns tant warping function o n co mpact ma nifolds. F rom Corolla ry 1.4 only non-K¨ ahler ones are pos s ible. Indeed in [12] w ar p e d pr o duct Einstein metrics are constr ucted on a clas s of S 2 bundles ov er K¨ ahler-Einstein bas es warped with S m for m ≥ 2 , giving compact nontrivial quasi-Einstein metrics for pos itive inte- gers n ≥ 4, m ≥ 2. When m = 1, there are no nontrivial quasi-Eins tein metrics on compact manifolds, see Remark 3.5 and Pr op osition 2.1. When n = 3 , m ≥ 2 it remains op en. In Section 4 w e also g ive a c haracter iz a tion of quasi-Eins tein metrics with finite m whic h are Eins tein at the same time. 2. W arped Product Einstein Metrics In this se c tion we show that when m is a p ositive integer the quasi- Einstein metrics (1.2) cor r esp ond to some warp ed pro duct Einstein metrics, mainly due to the work o f [10]. Recall that given tw o Riemannian manifo lds ( M n , g M ), ( F m , g F ) a nd a pos itive smo oth function u on M , the warp ed pro duct metric on M × F is defined by (2.3) g = g M + u 2 g F . RIGIDITY OF QUASI-EINSTEIN ME TRICS 3 W e denote it as M × u F . W arp ed pr o duct is very useful in co nstructing v a rious metrics. When 0 < m < ∞ , conside r u = e − f m . Then we hav e ∇ u = − 1 m e − f m ∇ f , m u Hess u = − Hess f + 1 m d f ⊗ d f . Therefore (1.2) can b e r ewritten as (2.4) Ric − m u Hess u = λg . Hence we can use equatio n (2.4) to study (1.2) when m is finite a nd vice verse. T aking trace of (2.4 ) w e hav e (2.5) ∆ u = u m ( R − λn ) . Since u > 0 this immediately gives the following r esult which is similar to the m = ∞ (soliton) c a se. Prop ositio n 2. 1. A c omp act quasi-Einstein metric with c onstant sc alar curvatur e is trivia l. In [10] it is sho wn that a Riemannian manifold (M,g) satisfies (2.4) if and only if the w a r p e d pro duct metr ic M × u F m is Einstein, where F m is an m -dimensional Einstein manifold with E ins tein constant µ sa tisfies µ = u ∆ u + ( m − 1) |∇ u | 2 + λu 2 . (In [10] it is only stated for co mpact Riemannian manifold, while compactnes s is redundant. Also the Laplacian there and her e ha ve differen t sign.) Therefore we hav e the following nice character iz ation of the quasi- Einstein metrics as the bas e metrics of warped pro duct E instein metrics. Theorem 2.2. ( M , g ) satisfies the quasi-Einstein e quation (1.2) if and only if the warp e d pr o duct metric M × e − f m F m is Einst ein, wher e F m is an m - dimensional Einstein manifo ld with Einstein c onstant µ satisfying (2.6) µe 2 m f = λ − 1 m  ∆ f − |∇ f | 2  . 3. Formulas and Rigidity for Quasi-Einstein Metrics In this section we gener alize the calc ula tions in [14] fo r Ricci solito ns to the metrics satisfying the qua si-Einstein equation (1.2). Recall the following general formulas, s ee e.g. [14, Lemma 2.1 ] for a pro of. Lemma 3.1. F or a function f in a R iemannian manifold (3.7) 2(div Hess f )( ∇ f ) = 1 2 ∆ |∇ f | 2 − | Hess f | 2 + Ric( ∇ f , ∇ f ) + h∇ f , ∇ ∆ f i , (3.8) div ∇∇ f = Ric ∇ f + ∇ ∆ f . 4 JEF FREY CASE, YU-JE N SHU, AND GUOF ANG WE I The trace for m of (1.2) (3.9) R + ∆ f − 1 m |∇ f | 2 = λn, where R is the sca lar curv ature, will b e used later. Using these formulas and the contracted seco nd Bianchi identit y (3.10) ∇ R = 2 div Ric , we ca n show the following formulas for quasi-Einstein metr ics, which g eneralize some of the formulas in Section 2 of [14]. Lemma 3.2. If Ric m f = λg , then 1 2 ∆ |∇ f | 2 = | Hess f | 2 − Ric( ∇ f , ∇ f ) + 2 m |∇ f | 2 ∆ f , (3.11) 1 2 ∇ R = m − 1 m Ric( ∇ f ) + 1 m ( R − ( n − 1) λ ) ∇ f , (3.12) 1 2 ∆ R − m + 2 2 m ∇ ∇ f R = m − 1 m tr (Ric ◦ ( λI − Ric)) − 1 m ( R − nλ )( R − ( n − 1) λ ) = − m − 1 m     Ric − 1 n Rg     2 − m + n − 1 mn ( R − nλ )( R − n ( n − 1) m + n − 1 λ ) . (3.13) Pr o of. F rom (3.7) w e hav e (3.14) 1 2 ∆ |∇ f | 2 = 2(div Hess f )( ∇ f ) + | Hes s f | 2 − Ric( ∇ f , ∇ f ) − h∇ f , ∇ ∆ f i . By taking the divergence of (1.2), we ha ve (3.15) div Ric + div Hess f − 1 m ∆ f d f − 1 m ( ∇ ∇ f ∇ f ) ∗ = 0 , where X ∗ is the dual 1- form of the vector field X . Using (3.10) we get (3.16) 2 div Hess f ( ∇ f ) = − h∇ R, ∇ f i + 2 m ∆ f |∇ f | 2 + 2 m Hess f ( ∇ f , ∇ f ) . Now taking the co v ar iant deriv a tive of (3.9) y ields (3.17) ∇ R + ∇ ∆ f − 1 m ∇|∇ f | 2 = 0 . Plug this into (3.16) and then plug (3 .16) into (3.1 4) we get 1 2 ∆ |∇ f | 2 = h∇ ∆ f − 1 m ∇|∇ f | 2 , ∇ f i + 2 m ∆ f |∇ f | 2 + 2 m Hess f ( ∇ f , ∇ f ) + | Hess f | 2 − Ric( ∇ f , ∇ f ) − h∇ f , ∇ ∆ f i = | Hess f | 2 − Ric( ∇ f , ∇ f ) + 2 m |∇ f | 2 ∆ f , which is (3.1 1). RIGIDITY OF QUASI-EINSTEIN ME TRICS 5 F or (3.12), us ing (3.10), (3.15), (3.8), a nd (3.17), we get ∇ R = 2 div Ric = − 2 div Hess f + 2 m ∆ f ∇ f + 2 m ∇ ∇ f ∇ f = − 2 Ric( ∇ f ) − 2 ∇ ∆ f + 2 m ∆ f ∇ f + 2 m ∇ ∇ f ∇ f = − 2 Ric( ∇ f ) + 2 ∇ R − 2 m ∇|∇ f | 2 + 2 m ∆ f ∇ f + 2 m ∇ ∇ f ∇ f . Solving for ∇ R and noting that ∇|∇ f | 2 = 2 ∇ ∇ f ∇ f , we get ∇ R = 2Ric( ∇ f ) − 2 m ∆ f ∇ f + 2 m ∇ ∇ f ∇ f . F rom (1.2), we ha ve that ∇ ∇ f ∇ f =  λ + 1 m |∇ f | 2  ∇ f − Ric( ∇ f ) , so by using this substitution and (3.9) for ∆ f , w e arr ive a t (3.12), 1 2 ∇ R = m − 1 m Ric( ∇ f ) + 1 m ( R − ( n − 1) λ ) ∇ f . T aking the divergent of the abov e equation w e hav e (3.18) 1 2 ∆ R = m − 1 m div (Ric( ∇ f )) + 1 m div (( R − ( n − 1) λ ) ∇ f ) . Now div (Ric( ∇ f )) = h div Ric , ∇ f i + tr (Ric ◦ Hess f ) = h 1 2 ∇ R, ∇ f i + tr  Ric ◦  1 m d f ⊗ d f + λg − Ric  = h 1 2 ∇ R, ∇ f i + 1 m Ric( ∇ f , ∇ f ) + tr (Ric ◦ ( λg − Ric)) = h 1 2 ∇ R, ∇ f i + 1 m − 1  h 1 2 ∇ R, ∇ f i − 1 m ( R − ( n − 1) λ ) |∇ f | 2  (3.19) +tr (Ric ◦ ( λg − Ric)) , where the last equatio n comes from (3.12). Also (3.20) div (( R − ( n − 1) λ ) ∇ f ) = ( R − ( n − 1) λ ) ∆ f + h∇ R, ∇ f i . Plugging (3.1 9) and (3.20) into (3.1 8) and using (3 .9) we a rrive at 1 2 ∆ R − m + 2 2 m ∇ ∇ f R = m − 1 m tr (Ric ◦ ( λI − Ric)) − 1 m ( R − nλ )( R − ( n − 1) λ ) . Let λ i be the eigenv alues of the Ricc i tensor, we get tr (Ric ◦ ( λI − Ric)) = X λ i ( λ − λ i ) = −     Ric − 1 n Rg     2 + R  λ − 1 n R  , which yields (3.13 ).  6 JEF FREY CASE, YU-JE N SHU, AND GUOF ANG WE I As in [14], these form ula s give imp ortant information ab out q uasi-Einstein met- rics. Combining the first equa tion (3.11) in Lemma 3.2 with the maxima l principle we hav e Prop ositio n 3.3. If a c omp act R iemannian manifold satisfying (1.2) and Ric( ∇ f , ∇ f ) ≤ 2 m |∇ f | 2 ∆ f then t he fun ction f is c onstant so it is Einstein. Equation (3.12) g ives Prop ositio n 3.4. When m 6 = 1 , a quasi-Einstein metr ic has c onstant sc alar cur- vatur e if and only if Ric( ∇ f ) = − 1 m − 1 ( R − ( n − 1) λ ) ∇ f . Remark 3.5. When m = 1 , the c onstant µ in (2.6) is zer o, c ombining with (3.9) , we get R = ( n − 1) λ . The sc alar curvatur e is always c onstant. Equation (3.13) g ives the following results. Prop ositio n 3.6. If a Rie mannian manifol d M satisfies (1.2) with m ≥ 1 and a) λ > 0 and M is c omp act then t he sc alar curvatu r e is b ounde d b elow by (3.21) R ≥ n ( n − 1) m + n − 1 λ. Equality holds if and only if m = 1 . b) λ = 0 , the sc alar curvature is c onstant and m > 1 , then M is Ric ci flat. c) λ < 0 and the sc alar curvatur e is c onstant, then nλ ≤ R ≤ n ( n − 1) m + n − 1 λ and whe n m > 1 , R e quals either of the extr eme values iff M is Einstein. Remark 3.7. When m is finite, a m anifold with quasi-Einstein metric and λ > 0 is automatic al ly c omp act [16] . Remark 3.8. L et m = ∞ , we r e c over the wel l know r esult [8] that c omp act shrink- ing Ric ci soliton has p ositive sc alar curvature , and some r esu lts in [14] ab out gr a- dient R ic ci solitons with c onstant sc alar curvature . Pr o of. a ) Since M is c o mpact, applying the equa tion (3.13) to a minimal po int of R , we have − m + n − 1 mn ( R min − nλ )( R min − n ( n − 1) m + n − 1 λ ) ≥ m − 1 m     Ric − 1 n Rg     2 ≥ 0 . So n ( n − 1) m + n − 1 λ ≤ R min ≤ nλ which gives (3.21). b) c) Since R is constant, from (3.1 3) − m + n − 1 mn ( R − nλ )( R − n ( n − 1) m + n − 1 λ ) = m − 1 m     Ric − 1 n Rg     2 ≥ 0 . RIGIDITY OF QUASI-EINSTEIN ME TRICS 7 So if λ = 0 , m > 1, then Ric = 1 n Rg and R = 0 , th us it is Ricci flat. If λ < 0, R ∈ [ nλ, n ( n − 1) m + n − 1 λ ].  4. Two Dimensional Quasi-Einstein Metrics First we recall a characterizatio n o f warp ed pro duct metrics found in [4] (see also [15]). Theorem 4.1 (Cheeger- C o lding) . A Riemannian manifold ( M n , g ) is a warp e d pr o duct ( a, b ) × u N n − 1 if and only if ther e is a n ontrivial funct ion h such that Hess h = k g for s ome function k : M → R . ( u = h ′ up to a multiplic ative c onstant) F rom this we can give a characterization of quasi-E instein metrics which are Einstein. Prop ositio n 4.2. A c omplete finite m quasi-Einstein metric ( M n , g , u ) is Einstein if and only if u is c onst ant or M is diffe omorphic to R n with the warp e d pr o duct structur e R × a − 1 e ar N n − 1 , whe r e N n − 1 is R ic ci flat, a is a c onstant (se e b elow for its value). Pr o of. If g is Eins tein, then Ric = µg for some cons tant µ . F rom (2.4) we have (4.22) Hess u = µ − λ m u g for so me u > 0 . If M is compact, then u (th us f ) is consta n t. So if u is not constant, then M is noncompact and λ ≤ 0 , µ ≤ 0 a nd µ > λ . So λ < 0 , λ < µ ≤ 0 and u is a strictly conv ex function. Ther efore M n is diffeomorphic to R n . By (4.2 2) and Theo rem 4 .1, u = ce √ µ − λ m r , where c is some c o nstant. And M is R × N n − 1 with the warp ed pro duct metric g = dr 2 + a − 2 e 2 ar g 0 , where a = q µ − λ m . Since g is Einstein w e get µ = − ( n − 1) µ − λ m < 0 a nd g 0 is Ricci flat.  Remark 4.3. The T aub-NUT metric [7 ] is a Ric ci flat metric on R 4 which is n ot flat. Since a 2- dimens io nal Riemannia n ma nifold s atisfies Ric = R 2 g , we get an imme- diate corolla r y of Theorem 4.1. Corollary 4.4. A two dimensional quasi-Einstein metric 2.4 is a warp e d pr o duct metric. Now we will prov e Theorem 1.2. Pr o of. Since M is compact, by [10], we only need to prove the theorem when λ > 0. F rom (3.21) we hav e (4.23) R ≥ 2 m + 1 λ. So up to a cov er we may assume M is diffeomorphic to S 2 . Since M is 2-dimensional, we hav e Ric = R 2 g . Th us (3.12) b ecomes (4.24) ∇ R = m + 1 m  R − 2 m + 1 λ  ∇ f . 8 JEF FREY CASE, YU-JE N SHU, AND GUOF ANG WE I Now let u = e − f m , then fro m (2.4) Hess u = u m  R 2 − λ  g . In particular , ∇ u is conformal. By the Kazdan-W arner ident ity [9], w e hav e Z M h∇ R, ∇ u i dV = 0 . Thu s − 1 m Z h∇ R, ∇ f i e − f m dV = 0 . Using (4.24), sinc e R ≥ 2 m +1 λ , we get ∇ f = ∇ R = 0.  5. K ¨ ahler Quasi-Einstein Metrics There ar e many no nt rivial examples of shrinking K¨ ahler-Ricci solitons [2, 17]. In co nt rast, w e will sho w here that K¨ ahler quasi-Einstein metr ics with finite m are very rigid. Pr o of of The or em 1.3 First, since on a K¨ ahle r m anifold, the metr ic and Ricci tensor a re b o th compatible with the complex structure J , from (2.4) we have Hess u ( J U , J V ) = Hess u ( U, V ) for all vector fields U, V . Tha t implies (5.25) J ∇ X ∇ u = ∇ J X ∇ u, and the φ defined by φ ( U, V ) = Hess u ( J U, V ) is an (1,1)-form. Note tha t 2Hess u = L ∇ u g . By (5.25) L ∇ u J X = J L ∇ u X , so 2 φ = L ∇ u ω , where ω is the K¨ ahler form. Since L X d = dL X and ω is closed we have φ is clo sed. F urthermore, since the Ricci form is clo s ed, from (2.4) we g e t that the (1,1)-for m φ u is also closed, so du ∧ φ = 0. Now ( du ∧ φ )( U, V , W ) = U u · g ( ∇ J V ∇ u, W ) + V u · g ( ∇ J W ∇ u, U ) + W u · g ( ∇ J U ∇ u, V ) . Let U, V ⊥ ∇ u, W = ∇ u , we hav e Hess u ( J U , V ) = 0 for a ll U, V ⊥ ∇ u . Hence (5.26) ∇ X ∇ u k J ∇ u for all X ⊥ ∇ u. Let U = ∇ u, V = J ∇ u, W ⊥ ∇ u we g et ∇ ∇ u ∇ u k∇ u . Now we consider the 2-dimensional distribution T 1 that is spanned b y ∇ u and J ∇ u at tho se p oints wher e ∇ u is nonz e r o. W e will show that T 1 = Span {∇ u, J ∇ u } is inv ar iant under parallel trans p or t, i.e. if γ is a path in M , and U is a parallel field along γ , then (5.27) ∇ γ ′  g ( U, ∇ u ) ∇ u |∇ u | 2 + g ( U, J ∇ u ) J ∇ u |∇ u | 2  = 0 . Since the cov a riant deriv ative is linear in γ ′ , we ca n prov e this in three cas es: (1) when γ ′ ⊥∇ u, J ∇ u : so γ ′ ⊥∇ u and J γ ′ ⊥∇ u , by (5.26) ∇ γ ′ ∇ u = 0. Since the complex structur e J is parallel, ∇ γ ′ J ∇ u = J ∇ γ ′ ∇ u = 0. By as s ump- tion ∇ γ ′ U = 0, hence (5.2 7) follo ws. (2) when γ ′ = ∇ u : Using ∇ ∇ u ∇ u k∇ u , w e hav e RIGIDITY OF QUASI-EINSTEIN ME TRICS 9 ∇ γ ′  g ( U, ∇ u ) ∇ u |∇ u | 2 + g ( U, J ∇ u ) J ∇ u |∇ u | 2  = γ ′  g ( U, ∇ u ) |∇ u | 2  ∇ u + g ( U, ∇ u ) |∇ u | 2 ∇ γ ′ ∇ u + γ ′  g ( U, J ∇ u ) |∇ u | 2  J ∇ u + g ( U, J ∇ u ) |∇ u | 2 ∇ γ ′ J ∇ u = g ( U, ∇ ∇ u ∇ u ) |∇ u | 2 ∇ u − 2 g ( ∇ u, ∇ ∇ u ∇ u ) g ( U, ∇ u ) |∇ u | 4 ∇ u + g ( U, ∇ u ) |∇ u | 2 g ( ∇ u, ∇ ∇ u ∇ u ) |∇ u | 2 ∇ u + g ( U, ∇ ∇ u J ∇ u ) |∇ u | 2 J ∇ u − 2 g ( ∇ u, ∇ ∇ u ∇ u ) g ( U, J ∇ u ) |∇ u | 4 J ∇ u + g ( U, J ∇ u ) |∇ u | 2 g ( J ∇ u, ∇ ∇ u J ∇ u ) |∇ u | 2 J ∇ u = ∇ u |∇ u | 2  Hess u ( U, ∇ u ) − Hess u ( ∇ u, ∇ u ) |∇ u | 2 g ( U, ∇ u )  + J ∇ u |∇ u | 2  − Hess u ( J U, ∇ u ) + Hess u ( ∇ u, ∇ u ) |∇ u | 2 g ( J U, ∇ u )  = ∇ u |∇ u | 2 Hess u  U − g ( U, ∇ u ) |∇ u | 2 ∇ u, ∇ u  − J ∇ u |∇ u | 2 Hess u  J U − g ( J U, ∇ u ) |∇ u | 2 ∇ u, ∇ u  = 0 , where the last equality follows from (5.26) and that U − g ( U, ∇ u ) |∇ u | 2 ∇ u ⊥ ∇ u . (3) γ ′ = J ∇ u : Using J ∇ X ∇ u = ∇ J X ∇ u it reduce s to the previous case. Now we have an ortho gonal decomp osition of the tange nt bundle T M = T 1 ⊕ T ⊥ 1 that is inv a riant under par allel transp ort. By DeRham’s decomp os itio n theorem on a simply-co nnected manifold [11, Page 1 87], M is a Riemannian pro duct, and all the claims in the theo rem follow.  Pr o of of Cor ol lary 1.4 Since the manifold M is compact, by [10], we can assume λ > 0. Then, by [16], the universal cov er ˜ M is also co mpact. Now the r esult follows from Theorem 1.3 and 1.2.  References [1] A.L. Besse , Einstein manifolds , Springer-V erlag, 1987. [2] H. Cao , Existe nc e of gr adient K¨ ahler-R icci solitons , E l liptic and Parabolic Metho ds i n Ge- ometry , A K Peters, W ellesley (1996), pp. 1–16. [3] Cao, Huai-Dong, Ge ometry of R ic ci solitons. Chinese Ann. M ath. Ser. B 27 (2006), no. 2, 121–142 [4] J. Cheeger, T. 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In terscience Pub- lishers, a divisi on of John Wiley & Sons, New Y ork-Lond on 1963 xi+329 pp. 10 JEFFREY CASE, YU-JEN SHU, AND GUOF ANG WE I [12] L, H., P age, Don N., P op e, C. N. New inhomo ge ne ous Einstein metrics on spher e bund le s over Einstein-K hler manifolds. Phys. Lett. B 593 (2004), no. 1-4, 218–226. [13] L. Ni and N. W allac h, On a classific ation of the gr adient shrinking solitons , arxiv:ma th.DG/071 0.3194 . [14] P . Peterson and W. Wylie, Rigidity of R icci Solitons , arxiv:mat h.DG/0710 .0174v1 . [15] P . Pe terson and W. Wylie, On the classific ation of gr adient R ic ci solitons , arxiv:ma th.DG/071 2.1298v2 . [16] Z. Qian , Estimates for weighte d volumes and applic ations , Quart. J. Math. Oxford Ser. (2) 48, No. 190 (1997) pp. 235–242. [17] X. W ang, X. Zhu , K¨ ahler Ric ci solitons on toric manifolds with p ositive first Chern class , Adv ances in Mathematics 188, (2004), pp. 87–103. [18] G. W ei and W. Wylie, Comp arison Ge ometry for the Smo oth Metric Me asur e Sp ac es , ICCM, 2007. E-mail addr ess : casej@ma th.ucsb. edu E-mail addr ess : yjshu@ma th.ucsb. edu E-mail addr ess : wei@math .ucsb.ed u Dep ar tment of Ma thema tics, UCSB, Sant a Barb ara, CA 9 3106