MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU Abstact. Motivated ad ispied by the ecet wok o Coldig [5] ad Coldig-Miicozzi [6] we deive seveal amilies o mootoicity omulas o maiolds with oegative Baky-Emey Ricci cuvatue, extedig the omulas i [5, 6]. 1. Itoductio The Baky-Emey Ricci teso is a Ricci teso o smooth metic measue spaces, which ae Riemaia maiolds with measues coomal to the Riemaia measues. Fomally a smooth metic measue space is a tiple (M, g, e dvol g ), whee M is a complete -dimesioal Riemaia maiold with metic g, is a smooth eal valued uctio o M, ad dvol g is the Riemaia volume desity o M. These spaces occu atually as smooth collapsed limits o maiolds ude the measued Gomov-Hausdo covegece. The N-Baky-Emey Ricci teso is (1.1) Ric N Ric + Hess 1 d d o 0 N. N The pupoted dimesio o the space is elated to N, i.e. it is + N. Whe N 0, we assume is costat ad Ric N Ric, the usual Ricci cuvatue. Whe N is iiite, we deote Ric Ric Ric + Hess. Note that i N 1 N the Ric N1 Ric N so Ric N λg implies Ric λg. The Eistei equatio Ric λg (λ a costat) is exactly the gadiet Ricci solito equatio, which plays a impotat ole i the theoy o Ricci low. O the othe had, the equatio Ric N λg, o N a positive itege, coespods to waped poduct Eistei metic o M F N, whee F N is some N dimesioal e N Eistei maiold, see []. Recetly Coldig [5] ad Coldig-Miicozzi [6] itoduced some ew mootoicity omulas associated to positive Gee s uctio o the Laplacia. These omulas ae vey useul ad elated to othe kow mootoicity omulas, see [7]. I paticula, usig oe o the mootoicty omula Coldig-Miicozzi showed that o ay Ricci-lat maiold with Euclidea volume gowth, taget coes at iiity ae uique as log as oe taget coe has a smooth coss-sectio [8]. Ou pape is motivated ad ispied by these wok o Coldig ad Coldig-Miicozzi. 000 Mathematics Subject Classiicatio. Pimay 53C0. Key wods ad phases. Mootoicity Fomulas, Baky-Emey Ricci Cuvatue. Patially suppoted by NSFC Gat No. 1117159. Patially suppoted by NSF Gat # DMS-1105536. 1

BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU With espect to the measue e dvol the atual sel-adjoit -Laplacia is. Coside the positive Gee s uctio G(x 0, ) o the -Laplacia o (M, g, e dvol) (see Deiitio.1). Fo ay eal umbe k >, let b G 1 k. Fo β, l, p R, whe b is pope, we coside A β () 1 l b β+1 e, While A β V () p l b +β b b p e. () is well deied o all > 0, V () is oly well deied whe (1.) C(, k, p) ( )(k p) β(k ) > 0. See the poo o Lemma 4.1 o detail. Whe k l, β, p 0, these educe to A(), V () i [5]. Whe k l, p, these ae A β, V β i [6]. Fist we obtai the ollowig gadiet estimate o b. Popositio 1.1. I a smooth metic measue space (M, g, e dvol) ( 3) has Ric N 0, the o k + N, thee exists 0 > 0, such that o M \ B(x 0, 0 ), (1.3) b(y) C(, N, 0 ). Remak I [5, Theoem 3.1] Coldig obtaied the shap estimate that i Ric M 0 ( 3), the b 1 o 0, k i above. Fom (.) this ca ot be tue whe k > as b(y) as y x 0. Fo Ric 0, b(y) may ot be bouded as y, see Example 6.5. We pove may amilies o mootoicity omulas, which, besides ecoveig the oes i [5, 6] whe k l ad is costat, give some ew oes eve i this case. Fo example, whe N is iite, we have Theoem 1.. I M ( 3) has Ric N 0, the, o k + N, k l k, α 3k p l ad C(, k, p) > 0, (1.4) (A β αv ) () p 1 l b Hece i i additio β, the A β β b β { 4b p Hess b b } g + 4(β )b b e. αv is odeceasig i. See discussio i Sectio 5 o ull geeality. Note that whe N 0 (i.e. is costat ad Ric 0), ad p 0, β, k we get mootoicity o all l ; i the case whe β, l, this is the ist mootoicity omula i [5]. Theoem 1.3. I M ( 3) has Ric N 0, the o β, k l + N, (A β ) () 0 ad (V ) () 0 o p < + N βn. I act (1.5) (A β ) () β 4 k 3 b k b β Hess b b g e. b

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 3 Agai this educes to a omula i [5] whe k l ad is costat, which is used i [8] to show that o ay Ricci-lat maiold with Euclidea volume gowth, taget coes at iiity ae uique as log as oe taget coe has a smooth coss-sectio. Without ay assumptio o we also establish mootoicity omulas whe N is iiite. Fo example, Theoem 1.4. I M ( 4) has Ric 0, the o β, p 0, k 1, l 3 k 1, we have (Aβ ( 3 k 1)V ) () 0; o β, k 1, l 3 (k 1), 1 > 0, we have (A β ) ( ) (A β ) ( 1 ). Hee l > k. See Theoem 6.1 o moe geeal statemet. Whe k l, simila mootoicity is ot tue ay moe. I act o k l we show while o 3 seveal mootoity still holds o the Byat solitos o lage, it does ot hold whe 5. I act it is mootoe i the opposite diectio o lage, see Example 6.5. With some coditio o we still get seveal mootoicity, see e.g. Coollay 6.4. As i [6], oe ca use the tem Hess b b g to allow smalle β (oe oly eeds β 1 1 1 istead o β, see the last pat o Sectio 5). The pape is ogaized as ollows. I the ext sectio, we discuss the existece ad the basic popeties o Gee s uctio o -Laplacia ad pove Popositio 1.1. I Sectio 3, usig the ollowig Boche omula (see e.g. [17]) (1.6) 1 u Hess u + u, ( u) + Ric ( u, u). we compute the -laplacia o (b q b β ), a key omula eeded o deivig the mootoicity omulas i Sectio 4. I Sectio 5, 6 we apply these omulas to the case whe Ric N 0 o N iite ad iiite espectively. As with may othe mootoicity omulas, we expect ou omulas will have ice applicatios, especially o quasi-eistei maiolds ad steady gadiet Ricci solitos. Ackowledgmets. The authos would like to thak Toby Coldig o his iteest ad ecouagemet, Toby Coldig ad Bill Miicozi o asweig ou questios o thei wok. This wok was doe while the ist ad thid authos wee visitig UCSB. They would like to thak UCSB o hospitality duig thei stay.. Gee s uctio Deiitio.1. Give a smooth metic measue space (M, g, e dvol), x 0 M, G G(x 0, ) is the Gee s uctio o the -Laplacia (with pole at x 0 ) i G δ x0. I [13, 10] it is show that o ay complete Riemaia maiold thee exists a symmetic Gee s uctio o the Laplacia. Same poo caies ove o - Laplacia. M is called -opaabolic i it has positive Gee s uctio. Whe 0 ad Ric M 0, the existece o positive Gee s uctio is well udestood [18], see also [11, 16]. Namely M is opaabolic i ad oly i 1 VolB(x 0,) d <, ad G 0 at iiity. Same esult also holds whe Ric N 0 (N iite) o Ric 0

4 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU ad is bouded. This ad othe existece esults will be studied i [1]. I thee we obseve that all otivial steady Ricci solitos ae -opaabolic. I this pape we assume M is -opaabolic ad G 0 at iiity so b is pope. Also, ea the pole (ate omalizatio), o 3, by [9], we have (.1) G(y) d (x 0, y)(1 + o(1)), G(y) ( )d 1 (x 0, y)(1 + o(1)), whee o(1) is a uctio with o(y) 0 as y x 0. Recall o k >, b G 1 k. The (.) b(y) d k (x0, y)(1 + o(1)), b(y) k d k k (x0, y)(1 + o(1)). To pove Popositio 1.1, we eed the ollowig two popositios o Ric N 0. Simila to the case whe Ric 0, oe has the ollowig Laplacia compaiso ad gadiet estimate o Ric N 0 [14, 15], see also [17, 1]. Let (x) d(x, x 0 ) be the distace uctio, the (.3) R +N + N. I u is a -hamoic uctio o B(x, R), the o B(x, R/), C( + N) (.4) log u. R Poo o Popositio 1.1. Fo ay y M \ {x 0 }, G is a smooth hamoic uctio o B(y, ) with (y) d(y, x 0 ). By (.4), log G C(k) Now log b 1 k log G. Theeoe o B(y, /). b (y) log b b C(k) k b. Hece (1.3) ollows i we show b C 1 (, N, 0 ), i.e. G C (, N, 0 ) k o some 0 > 0 o M \ B(x 0, 0 ). Fo ay ɛ > 0, we have o M \ {x 0 }, (G ɛ k ) ɛ ( k ) ɛ R k( k ) 0, whee the iequality ollows om the Laplacia compaiso (.3). Sice lim (G ɛ k ) 0. Also by (.1), thee exists 0 > 0 small such that G(y) 1 0 o B(x 0, 0 ). Take ɛ 1 k 0, we have (G 1 k 0 k ) 0 o B(x 0, 0 ). Theeoe by the maximum piciple we have G(y) 1 k 0 k whe (y) 0. Namely b ( k 0 ) 1 k. Lemma 3.1. Fo ay eal umbe β, 3. The -Laplacia o b ad b (3.5) b k 1 b, b (3.6) b β β(β + k )b β b. I paticula, (3.7) b k b.

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 5 Poo: Fo ay positive uctio v, we have [ ] (3.8) v β βv β 1 (β 1) v + v v Sice b k 0, this gives ( k 1) b b + b 0, amely (3.5). Combiig (3.8) ad (3.5) gives (3.6). The ollowig impotat omulas holds o ay positive -hamoic uctio G, ot just Gee s uctio. Popositio 3.. b β β 4b b β { Hess b + Ric ( b, b ) + (k ) b, b (3.9) +4(β )b b 4k b 4}. (b q b β ) β { 4 bq b β Hess b + Ric ( b, b ) + (k + q) b, b [ ] } 8q (3.10) +4(β )b b + (k + q) 4k b 4 β Poo: Sice the omulas ae i tems o the uctio b, we ist compute b. Applyig the Boche omula (1.6) to b ad usig (3.7), we have (3.11) 1 b Hess b + Ric ( b, b ) + b, ( b ) Hess b + Ric ( b, b ) + k b, b. Now we compute b. Sice b 4b b, 1 b ( b b + b b + b, b ) Combie this with (3.11), we have (3.1) The, 4k b 4 + b b + 4 b, ( b ). b b Hess b + Ric ( b, b ) + (k 4) b, b b β ( b ) β which is (3.9). 4k b 4. β b β b + β(β ) b β b β 4b b β { Hess b + Ric ( b, b ) + (k 4) b, b +4(β )b b 4k b 4},

6 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU Fo the secod oe, by the poduct omula o Laplacia ad usig (3.6), we get (b q b β ) b q b β + b β (b q ) + b β, b q Plug i (3.9) we obtai (3.10). Recall o l, β, p R, b q b β + q(q + k )b q b +β +βq b β b q b, b. 4. Mootoicity Fomulas A β () 1 l b β+1 e, V () p l b +β b b p e. As 0, we have the ollowig iomatio. Lemma 4.1. Let M be a smooth maiold with 3. Deote C(, k, l) (k l)( ) + ( k)β. The 0 i C(, k, l) > 0 (4.1) lim 0 Aβ () lim V 0 () (4.) ( k ) 1+β V ol( B1 (0))e (x0) i C(, k, l) 0 i C(, k, l) < 0 0 ( i C(, k, l) > 0 ) 1+β k C(,k,p) Vol( B 1(0))e (x0) i C(, k, l) 0 i C(, k, l) < 0 whee Vol( B 1 (0)) is the volume o the uit sphee i R. Poo: Fom (.), A β () 1 l ( k b 1+β e ) 1+β ( 1 l+ k (1+β)+ k ( 1)) (1 + o(1))e (x0) Vol( B 1 (0)), whee o(1) 0 as 0. Note that 1 l + k k (1 + β) + ( 1) (k l)( ) + ( k)β. This gives (4.1). Similaly, V () p l 0 ( k bs b 1+β b p ) 1+β p l 0 e s k k (1+β) p+ ( 1) ds(1 + o(1)) V ol( B 1 (0))e (x0) The itegal exists i the costat i (1.), C(, k, p) > 0, ad V () ( ) 1+β C(,k,l) (1 + o(1))v ol( B 1 (0))e (x0) C(, k, p) k,,

This gives (4.). Sice we have MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 7 V () p l (V ) () (p l) p l 1 0 0 bs bs b 1+β b p b 1+β b p e. e + p l (4.3) p l V () + 1 Aβ (). To id the deivative o A (), we use the ollowig omula. Lemma 4.. Fo a smooth uctio u : M \ x R, let I u () u b e. The o ay 0 < 0, (4.4) (4.5) I u() k 1 I u () + k 1 I u () + 0 b whee ν b b is the uit omal diectio. u, ν e b 1+β b p ( u) e + u, ν e, 0 This omula ca be deived usig the dieomophisms geeated by b b, see [4, Appedix]. Fo completeess, we give a simple poo usig just the diveget theoem ad the co-aea omula. Poo: Note that I u () I u ( 0 ) 0 b 0 u b e u b e 0 bs div(ue b) div(ue b). b Take deivative o this equatio both sides with espect to ad usig (3.5) gives I u div(ue b) b u, b e + ue b b b u, b e + ue k 1 b b ( u) e b + u, 0 b 0 b e + k 1 u b e. I ode to match the deivative o A β e with V, we wite A β () p l 1 b p b β+1 e. Applyig (4.5) to u b p b β, i C(, k, p) > 0, the itegal o b 0 goes to zeo as 0 0, ad we have

8 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU Coollay 4.3. (4.6) (A β ) () k l + p A β () + p 1 l b ( (b p b β ) ) e. As i [5] we deive a omula which will give the ist mootoicity. Theoem 4.4. Whe k >, C(, k, p) > 0, o ay α R, (A β αv ) () p 1 l + 1 ( (4.7) whee (4.8) b β b β 4b p λ 1 A β () + λ V { Hess b + Ric ( b, b ) + 4(β )b b } e ) (), λ 1 3k p l α, (4.9) λ (p + k)(k p) βk α(p l). ( Poo: Fom (4.6) we would like to compute (b p b β ) ) e. By (3.10), (b p b β ) { β b β Hess b 4b p + Ric ( b, b ) + (k p) b, b (4.10) [ ] 8 4p +4(β )b b + (k p) 4k b }. 4 β b To compute the thid tem, by Stokes theoem ad (3.6), b p β b β b, b e (4.11) b b p b p, ( b ) β/ e b p b b, b b β e (b p ) b β e p b 1 p b β+1 e (k p) b p b β+ e l p A β + (p k)l p V. I the above poo we assume p. By takig limit o (4.11) as p, we see (4.11) also holds o p. I the secod equality, we use C(, k, p) > 0 so the itegal o 0 is zeo. Combiig (4.11),(4.10), (4.6) ad (4.3), we have (A β αv ) () p 1 l b β b β 4b p + 3k p l α A β () + 1 b { Hess b + Ric ( b, b ) + 4(β )b b } e [(p + k)(k p) βk α(p l)] V ().

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 9 Lettig k l, α k p i (4.7) gives Coollay 4.5. Whe k >, C(, k, p) > 0, ad k l, we have (A β (k p )V ) () p 1 l β b β { b 4b p Hess b (4.1) +Ric ( b, b ) + 4(β )b b 4k b 4 }e. Followig is a omula which will give secod mootoicity omula. Theoem 4.6. Fo c, d R, let g() c ( d A β () ). The o 0 < 1 <, (4.13) whee g( ) g( 1 ) 1 b { β 4 bc+d l 1 b β Hess b + Ric ( b, b ) + λ 3 b 4 + 4(β )b b } e + λ 4 1 b b c+d l b, b β e, (4.14) (4.15) λ 3 4 (k + d l + c 1)(k + d l) 4k, β λ 4 3k l 3 + c + d. Poo: Fom (4.4) with u b β, we have (A β ) () 1 l b β, ν e + (k l) l Hece g() ad Sice ad b 1+β e. b c+d+1 l b β, ν e + (k + d l) b c+d l b 1+β e, g( ) g( 1 ) div(b c+d+1 l e b β ) div(b c+d+1 l e b β ) 1 b +(k + d l) div(b c+d l b β e b). 1 b (c + d + 1 l)b c+d l b, b β e + b c+d+1 l ( b β )e div(b c+d l b β e b) (c + d l)b c+d 1 l b +β e + b c+d l b β, b e + b c+d l b β ( b)e,

10 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU pluggig b β ad b with (3.9) ad (3.5) to the above, we get g( ) g( 1 ) 1 b { β 4 bc+d 1 l b β [ Hess b + Ric ( b, b ) 4k b 4 + 4(β )b b ] } e + (k 3 + c + d l)b c+d l b, b β e +(k + d l) b c+d l [ (c + d l + k 1)b 1 b +β + b, b β ] e. 1 b This is (4.13) ate goupig. 5. Mootoicity o Ric N 0 Fom Theoem 4.4 ad Theoem 4.6, i Ric N 0, we get may amilies o mootoicity quatities. Sice Ric ( b, b ) Ric N ( b, b ) + b, N ad (5.1) Hess b Hess b b g + ( b ), we have Hess b + Ric ( b, b ) ( b ) + b, N ( b ) + N 4k + N b 4. Hee we used the basic iequality a p + b q (a+b) p+q. Theeoe, by Theoem 4.4, (A β αv ) () 0 i β, λ 1 0 ad λ + βk +N 0. Thee ae may solutios to these. Fo example, i we let λ 1 0, amely α 3k p l, ad k + N, the λ + βk +N 0 whe k l k,which gives Theoem 1.. Lettig k l + N i Theoem 1., we get Coollay 5.1. I M ( 3) has Ric N 0, the, o k l + N, β, p < + N βn, ad 0 < 1 <, (A β ( + N p )V )( ) (A β ( + N p )V )( 1 ). Similaly, by Theoem 4.6, g() c ( d A β () ) is odeceasig i β, λ3 + 4k +N 0 ad λ 4 0. Agai thee ae may solutios. λ 4 0 equies that c + d 3 3k + l. Whe c k 1, d l k + o c 3 3k + l, d 0, ad k l + N, we have Popositio 5.. I M ( 3) has Ric N 1 b 0, the o 0 < 1 <, k l +N, k 1 ( k A β ) ( ) 1 k 1 ( k A β ) ( 1 ) { } β 4 b k b β Hess b b g + 4(β )b b e,

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 11 ad 3 k (A β ) ( ) 1 3 k (A β ) ( 1 ) { β 4 b k b β Hess b b g (5.) 1 b + 4(β )b b } e. Agai, whe N 0, β, these ae the secod ad thid mootoicity omula i [5]. Now we pove Theoem 1.3 which we estate it hee. Theoem 5.3. I M ( 3) has Ric N 0, the o β, k l + N, (A β ) () 0 ad (V ) () 0 o p < + N βn. I act (5.3) (A β ) () β 4 k 3 b k b β Hess b b g e. b Poo: Fist we show A β () is bouded as. By (1.3), b(y) C(, N, 0) o M \ B(x 0, 0 ). Hece o 0, A β () 1 k b 1+β e C(, N, 0, β) 1 k Deie h() 1 k b e, by (4.4) h () 1 k 1, ν e 0. b e. By (4.1), lim 0 h() k Vol( B 1(0))e (x0). Theeoe (5.4) A β () C(, N, 0, β)e (x0) o 0. Next we pove that (A β ) () 0. By (5.), o 0 < 1 < 3 k (A β ) ( ) 3 k 1 (A β ) ( 1 ). I thee is some > 0 such that (A β ) ( ) > 0, the o all, ( (A β ) k 3 ) () (A β ) ( ) (A β ) ( ) > 0. Namely A β () as, cotadictig to (5.4). To show (V ) () 0, ote that by Coollay 4.1, (A β ( + N p ) () 0. Hece )V ( + N p )(V ) () (A β ) () 0. Now (5.3) ollows om (5.) sice the act that (A β ) () 0 ad A β () is bouded as imply thee ae sequece j such that 3 k j (A β ) ( j ) 0.

1 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU As i [6], oe ca decompose Hess b b g uthe moe to allow smalle β. Deote (5.5) B Hess b b g, the tace ee symmetic biliea om, whee g is the Riemaia metic. Oe ca decompose B ito the omal ad tagetial compoets. Let B 0, g 0 be the estictio o B, g to the level set o b, deie B(ν) the vecto such that B(ν), v B(ν, v), ad B(ν) T its tagetial compoet, so B(ν) (B(ν, ν)) + B(ν) T. The (5.6) B (B(ν, ν)) + B(ν) T + B 0 B(ν) + B(ν) T + (tb 0) 1 + B 0 tb 0 1 g 0 1 B(ν) + B(ν) T + 1 B 0 tb 0 1 g 0. Hee we used the act that B is tace ee so tb 0 B(ν, ν). Now Lemma 5.4. (5.7) B(ν) 4b b + λ + 4λb b, ν, whee λ b b Poo: Sice we have (1 k ) b 1 b,. Hess b bhess b + b b, B( b) bhess b( b) + b b b b b b + λ b, ad B(ν) b b + λν. Hece B(ν) 4b b + λ + 4λb b, ν. Theeoe we have Remak Whe λ 0 (as i the case i [6]) o b, ν 0, B 4 1 b b. We also get all above mootoicity o β 1 1 1 istead o β. 6. Mootoicity o Ric 0 Whe oe oly assumes Ric 0, oe does ot have the exta tem b, combie with Hess b to get ( b ). Fist we study the mootoicity without usig Hess b. Iteestigly we still get may mootoic omulas. Theoem 6.1. I M ( 3) has Ric 0, the o β, k, p such that C(, k, p) > 0, (k ) 4βk 0, l 1 l l, whee l 1 ad l ae the solutios o l + ( 3k)l + k k + βk 0, ad α 3k p l, we have (A β αv ) () 0. N to

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 13 Poo: Fom (4.4), we oly eed to make sue λ 1 0, λ 0. By the choice o α we have λ 1 0 ad λ 0 i ad oly i l + ( 3k)l + k k + βk 0. This has eal solutio whe (k ) 4βk 0. Whe β, p 0, 4, the o ay k 1, l 3 k 1 is betwee the l 1, l occued above, which is the ist pat o Theoem 1.4. Similaly, by Theoem 4.6, i β, c d 0, l 3 (k 1), k 1, the λ 4 0, λ 3 > 0, we get the secod statemet i Theoem 1.4. To get mootoicity o Ric 0 whe k l, we eed to use the Hess b tem. Hee ae some omulas whe k l, which ecove the omulas i [6] whe is costat ad p. Theoem 6.. Whe k l, p <, we have (6.1) (A β ( p )V ) () p 1 β b β { 4b p B + Ric ( b, b ) whee B is give i (5.5). Poo: Fom (4.1) (A β b +4(β )b b + 4 b b, + b, ( p )V ) () p 1 b } e, β b β { Hess b 4b p +Ric ( b, b ) + 4(β )b b 4 b 4 }e. By (5.1) ad (3.7), Hess b B + ( b + b, ) B + 4 b 4 + 4 b b, + b,. Plug this ito above gives (6.1). Similaly, lettig k l, d 0, c 3 i Theoem 4.6 gives Theoem 6.3. Whe k l, o 0 < 1 <, 3 (A β ) ( ) 1 3 (A β ) ( 1 ) { β 4 b b β B + Ric ( b, b ) 1 b +4(β )b b + 4 b b, + b, Coollay 6.4. I Ric 0, β, k l, p <, ad (6.) B + 4 b b, + b, 0, the (A β ( p )V ) () 0 } e.

14 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU ad o 0 < 1 <, 3 (A β ) ( ) 3 1 (A β ) ( 1 ). Note (6.) holds i paticula i b, 0 o b, 4 b. I geeal B + 4 b b, + b, could be egative ad mootoicity i Coollay 6.4 could eve be evesed. We illustate this i the ollowig example. Example 6.5. Byat solito is a otatioally symmetic steady gadiet Ricci solito. It is R ( 3) with the metic (6.3) g d + φ() g S 1 whee, as, C 1 1/ φ() C 1/ ad φ () O( 1/ ) o some positive costat C. Ad the potetial uctio () + O(l ) as. See e.g. [3]. The o >> 0, B + 4 b b, + b, is positive whe 3 ad egative whe 5, showig that o all 1 big, 3 (A β ) ( ) 3 1 (A β ) ( 1 ) i 3, 3 (A β ) ( ) 3 1 (A β ) ( 1 ) i 5. Poo: The Gee s uctio G o the -Laplacia with pole at the oigi is G G(), with G + ( 1) φ φ G G 0. Hece G e φ 1, ad G() With b G 1, we have b b 1 G G, ad b b G G G G + b b e φ 1 ds, both go to 0 as. ( 1) φ φ ( 1)G ( )G. Combie this with (5.6) ad (5.7), we have B 1 B(ν) 1 b b 1 b, ν (bb 1 ) 1 bb ( (bb ) G ( 1) ( )G + 1 ) φ. φ Now Hece 4 b b, (bb ) b b (bb ) G G. B + 4 b b, + b, ( (bb ) [( 1) G ( )G + 1 φ φ ) ] G G + ( ).

MONOTONICITY FORMULAS FOR BAKRY-EMERY RICCI CURVATURE 15 Sice lim () 1, lim φ we see that [ ( ( 1) lim φ 0 ad usig L Hospital s ule G lim G lim G G lim ( + (1 ) φ φ ) 1, G ( )G + 1 φ φ which is positive whe 3 ad egative whe 5. Now the last claim ollows om Theoem 6.3. ) ] G G + ( ) + 4 ( ), Reeeces [1] X. Dai, C.J. Sug, J. Wag ad G. Wei, i pepaatio. [] J. Case, Y. Shu, G. Wei, Rigidity o quasi-eistei metics, Dieetial Geom. Appl. 9 (011), o. 1, 93-100. [3] Chow, Beett; Chu, Su-Chi; Glickestei, David; Guethe, Chistie; Isebeg, James; Ivey, Tom; Kop, Da; Lu, Peg; Luo, Feg; Ni, Lei The Ricci low: techiques ad applicatios. Pat III. Geometic-aalytic aspects. Mathematical Suveys ad Moogaphs, 163. Ameica Mathematical Society, Povidece, RI, 010. [4] T.H. Coldig ad W.P. Miicozzi II, Hamoic uctios with polyomial gowth, Jou. Di. Geom. vol 45 (1997) 1-77. [5] T.H.Coldig, New mootoicity omulas o Ricci cuvatue ad applicatios; I, Acta Math. 09 (01), o., 9-63. [6] T.H. Coldig, W.P.Miicozzi, Ricci cuvatue ad mootoicity o hamoic uctios, axiv:109.4669. [7] T.H. Coldig, W.P.Miicozzi, Mootoicity ad its aalytic ad geometic implicatios, PNAS (01), i pess, axiv.og/abs/105.6768. [8] T.H. Coldig, W.P.Miicozzi, O uiqueess o taget coes o Eistei maiolds, axiv:106.499. [9] D.Gilbag, J.Sei, O isolated sigulaities o solutios o secod ode elliptic dieetial equatios, J.Aalyse Math.4 (1956) 309-340. [10] P. Li ad L. F. Tam, Symmetic Gee s uctios o complete maiolds, Ame. J. Math. 109 (1987), 119-1154. [11] P. Li, S.T. Yau, O the paabolic keel o the Schodige opeato, Acta Math. 156 (1986), 153-01. [1] X. Li, Liouville theoems o symmetic di?usio opeatos o complete Riemaia maiolds. J. Math. Pues Appl. (9) 84 (005), o. 10, 1951361. [13] M. Malgage, Existece et appoximatio des solutios des équatios aux deivées patielles et des équatios de covolutio, A. Ist. Fouie 6 (1955), 71-355. [14] Z.M. Qia, Gadiet estimates ad heat keel estimate. Poc. Roy. Soc. Edibugh Sect. A 15 (1995), o. 5, 975-990. [15] Z.M. Qia, Estimates o weighted volumes ad applicatios. Quat. J. Math. Oxod Se. () 48 (1997), o. 190, 35-4. [16] C.J. Sug, A ote o the existece o positive Gee s uctios, J. o uctioal Aal., 156 (1998), 199-07. [17] G. Wei ad W. Wylie, Compaiso Geomety o the Smooth Metic Measue Spaces, Poceedigs o the 4th Iteatioal Cogess o Chiese Mathematicias, Hagzhou, Chia, 007, Vol. II 191-0. [18] N.Vaopoulos, Potetial theoy ad diusio o Riemaia maiolds, I: Coeece o Hamoic aalysis i hoo o Atoi Zygmud, Vol. I, II(Chicago, Ill.,1981) E-mail addess: bysog@mail.ccu.edu.c School o Mathematics ad Statistics, Cetal Chia Nomal Uivesity, Wuha 430079, Chia E-mail addess: wei@math.ucsb.edu

16 BINGYU SONG, GUOFANG WEI, AND GUOQIANG WU Depatmet o Mathematics, UCSB, Sata Babaa, CA 93106 E-mail addess: cumtwgq@mail.ustc.edu.c Depatmet o Mathematics, Uivesity o Sciece ad Techology o Chia