Pacific Journal of Mathematics

Transcription

1 Pacfc Jornal of Mahemacs GRADIENT ESTIMATES FOR SOLUTIONS OF THE HEAT EQUATION UNDER RICCI FLOW SHIPING LIU Volme 43 No. 1 November 009

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3 PACIFIC JOURNAL OF MATHEMATICS Vol. 43, No. 1, 009 GRADIENT ESTIMATES FOR SOLUTIONS OF THE HEAT EQUATION UNDER RICCI FLOW SHIPING LIU We esablsh frs order graden esmaes for posve solons of he hea eqaons on complee noncompac or closed Remannan manfolds nder Rcc flows. These esmaes mprove Genher s resls by weakenng he crvare consrans. We also oban a resl for arbrary solons on closed manfolds nder Rcc flows. As applcaons, we derve Harnackype neqales and second order graden esmaes for posve solons of he hea eqaons nder Rcc flow. The resls n hs paper can be consdered as generalzng he esmaes of L Ya and J. Y. L o he Rcc flow seng. 1. Inrodcon In hs paper, we manly generalze L and Ya s [1986] and L s [1991] graden esmaes o posve solons of he hea eqaon nder Rcc flow. The Rcc flow, 1-1 g j = Rc j, was nrodced by Hamlon [198] o sdy he Poncaré conjecre on compac hree manfolds wh posve Rcc crvare. Snce hen, n he seres [1995; 1997; 1999], Hamlon creaed a well-developed heory of Rcc flow as an approach o he Poncaré conjecre and he geomerzaon conjecre. In [00; 003], Perelman brogh n new deas and compleed he so-called Hamlon program. Graden esmaes for solons of he hea eqaon are very powerfl ools n analyss, as shown for example n [L 1991; L and Ya 1986]. Perelman [00] acally showed a graden esmae for he fndamenal solon of he conjgae hea eqaon, R + = 0, nder Rcc flow on a closed Remannan manfold M, where R s he scalar crvare. Namely, le be he fndamenal solon of he eqaon above n M [0, T, MSC000: prmary 58J35, 53C44; secondary 35K55, 53C1. Keywords: graden esmae, Rcc flow, hea eqaon, Harnack neqaly. 165

4 166 SHIPING LIU and le f be he fncon sch ha = 4πτ n/ e f wh τ = T. Then τ f f + R + f n 0 n M [0, T. Eqvalenly, τ R n τ + ln + n ln4πτ. τ τ Ths esmae s mporan n he proof of Perelman. Recenly, Kang and Zhang [008] esablshed a graden esmae ha works for all posve solons of he conjgae hea eqaon nder Rcc flow on a closed manfold. Here and hrogho we say ha a Remannan manfold s closed f s compac who bondary. As an mmedae conseqence, hey ge a Harnack-ype neqaly. By spposng a lower bond on he Rcc crvare, Zhang [006] esablshed local graden esmaes for posve solons of he hea eqaon nder he backward Rcc flow g j = Rc j on a closed Remannan manfold. Under sronger crvare consrans, Genher [00] had already esablshed graden esmaes for posve solons of he hea eqaon nder Rcc flow on a closed manfold. Usng hs resl, she derved a Harnack-ype neqaly and fond a lower bond for he hea kernel nder Rcc flow. Here hea kernel means he fndamenal solon of he hea eqaon nder Rcc flow, whose exsence and basc properes Genher also proved. We weaken her crvare consrans n Secon sng he mehod of L and Ya [1986]. We also ge correspondng esmaes for complee noncompac manfolds nder Rcc flows. All of hese resls are generalzaons of L and Ya s graden esmaes. Ineresng n her own rgh, hgher order graden esmaes for hea kernels on complee noncompac Remannan manfolds nder Rcc flows are also closely relaed wh he bondedness of he Resz ransform and he Sobolev neqaly. Zhang [007] fond ha he noncollapsng resl, whch s crcal n Perelman s proof of he Poncaré conjecre, follows mmedaely from he Sobolev neqaly nder Rcc flow. L [1994] sed he bondedness of he Resz ransform o prove he Sobolev neqaly on Remannan manfold wh some consrans, so s naral o ry o prove wh a smlar mehod he Sobolev neqaly nder Rcc flow. In ha mehod, an mporan sep, compleed [L 1991], s o prove an esmae for X Y Hx, y,, where Hx, y, s he hea kernel and X and Y are he graden operaor n he varables x and y. However, dffcles arse n sng hs mehod o ge he generalzaon of hs esmae nder Rcc flow, snce n hs case he hea kernel has properes dfferen from wha had n he fxed merc case. As a conseqence of he frs order resls n Secon, n Secon 3 we ge he generalzaon of L s [1991] second order graden esmae for he posve solon x, of he hea eqaon nder Rcc flow. Le M be a complee noncompac Remannan manfold wh nal merc g0. Assme ha g evolves

5 GRADIENT ESTIMATES UNDER RICCI FLOW 167 by Eqaon 1-1 and s frs order covaran dervaves are bonded by k 1 and k. Then we have he noaon s defned n laer secons x, x, x, + α 5α x, x, x, Ck 1 + k /3 + 1/. In fac, he esmae for X Y Hx, y, and he second order graden esmae for x, are proved smlarly n [L 1991]. The man dfference s ha he laer doesn depend on he specal properes of he hea kernel. For closed Remannan manfolds nder Rcc flows, we ge a graden esmae for arbrary solons of he hea eqaon a he end of Secon. We wll se he followng noaons: We denoe by and he graden and Laplacan Belram operaor nder he merc g; by C a posve consan ha may change from lne o lne; by dx, y, he geodesc dsance beween x, y M nder g; and by ψr a C fncon on [0, +, sch ha { 1 f r [0, 1], 1- ψr = 0 f r [, +, and ψr 1, ψ r 0, ψ r C, ψ r C, ψr where C s an absole consan. When we say ha x, s a solon o he hea eqaon, we mean s a solon ha s C n x and C 1 n.. The frs order graden esmaes In hs secon, we prove he frs order graden esmaes. We wll denoe f x, f x, = f x, = for a fncon f on M [0, T ], where T s a posve consan. We gve a local verson graden esmae frs. Theorem 1. Le g be a solon o he Rcc flow on a Remannan manfold M n wh n for n some me nerval [0, T ], and sppose K 0 Rc K 1 for some posve consans K 0 and K 1 and all [0, T ]. Le M be complee nder he nal merc g0. Gven x 0 M and R > 0, le be a posve solon o he eqaon x, = 0 n he cbe Q R,T := {x, dx, x 0, R, 0 T }. Then for x, Q R,T, we have x, -1 α x, x, x, C K 1 + K R for any α > 1, where C depends on n and α only.

6 168 SHIPING LIU - More explcly, we have x, x, α x, x, nα + Cα R R K 0 + α α 1 for any α > 1, where C depends on n only. + nα3 α 1 K 0 + n 3/ α K 0 + K 1 + Cα K 1, As n he proof n [L and Ya 1986], le f = log, and le x, F = α x, = f α f. x, x, Lemma 1. Sppose M, g sasfes he hypoheses of Theorem 1. We have -3 F f F + n f f f α f αk 0 f α n K 0 + K 1. Proof. For a gven me, choose {x 1, x,..., x n } o be a normal coordnae sysem a a fxed pon. The sbscrps and j wll denoe covaran dervaves n he x and x j drecons. We wll compe a he fxed pon. By a drec compaon, we oban F =, j f j +, j f f j j +, j where we have sed he Rcc deny and he formla Rc j f f j α f + α, j -4 f = f Rc, Hess f. Rc j f j, On he oher hand, we have F = f α f +, j Rc j f f j + f f α f. Then nong ha f = f, we arrve a -5 F = f F + f j + α Rc j f j Becase Rc j n n s a real symmerc marx, we oban -6 K 0 K 1 Rc j K 1 + K 0 + α Rc f, f f α f. from K 0 Rc K 1. Applyng hose bonds and Yong s neqaly n he form Rc j f j α Rc j + 1 α f j,

7 GRADIENT ESTIMATES UNDER RICCI FLOW 169 we conclde -7 F f F + f j α n K 0 + K 1 The lemma s compleed wh he help of he neqaly, j f j f 1 n f = 1 n f f. αk 0 f f α f. Proof of Theorem 1. Bonded Rcc crvare mples ha g s nformly eqvalen o he nal merc g0 see [Chow e al. 006, Corollary 6.11], ha s, e K 1T g0 g e K 0T g0. By defnon, we know ha M, g s also complee for [0, T ]. Le dx, x0, ρx, ϕx, = ϕdx, x 0, = ψ = ψ, R R where ρx, = dx, x 0,. For he prpose of applyng he maxmm prncple, he argmen of [Calab 1958] allows s o assme ha he fncon ϕx,, wh sppor n Q R,T, s C a he maxmm pon. For any 0 < T 1 T, le x 1, 1 be he pon n Q R,T1 a whch ϕf acheves s maxmm vale. We can assme ha hs vale s posve, becase oherwse he proof s rval. Then a he pon x 1, 1, we have -8 ϕf = F ϕ + ϕ F = 0, ϕf 0, ϕf 0. Therefore, -9 0 ϕf = ϕf + ϕ F ϕ F + ϕ F. Usng he Laplacan comparson heorem, we have ϕ C R C R K0. By he evolon formla of he geodesc lengh nder Rcc flow see [Chow and Knopf 004], we calclae Fϕ = Fψ ρ 1 dρ R R d = Fψ ρ 1 RcS, S ds R R γ 1 Fψ ρ 1 R R K 1ρ Fψ ρ K 1 F C K 1, R

8 170 SHIPING LIU where γ 1 s he geodesc connecng x and x 0 nder he merc g 1, S s he n angen vecor o γ 1, and ds s he elemen of arc lengh. Sbsng he wo neqales above no -9 and sng -8, we oban C R C R K0 F C K 1 F + ϕ F. Applyng Lemma 1 o hs neqaly yelds C R C K0 F C 1 C K R 1 F ϕ f F + R n ϕ f f ϕ f α f αk 0 1 ϕ f 1 α n ϕk 1 + K 0. The followng compaon s almos he same as one n [L and Ya 1986]. Mlplyng hrogh by ϕ 1 and seng y = ϕ f and z = ϕ f, Eqaon -11 becomes C R C R K0 ϕf C C K 1 1 ϕf R 1 y1/ y αz + 1 n y z αk 0 1 y ϕ F 1 α n ϕ K 0 + K 1. Usng he neqaly ax bx b /4a for a, b > 0, one obans 1 C n y z R 1 y1/ y αz αk 0 1 y = 1 1 α 1 y n α y αz + αnk α 0 y+ α 1 n C α y R y1/ y αz 1 1 n α y αz α4 n K0 α 1 α n C y αz. α 1R Hence -1 becomes 1-13 nα ϕf ϕf 1 + C R 1 + C K0 R 1 + Cnα 1 α 1R + C K 1 1 nk 0 α 4 1 α α n ϕ K 1 + K 0 0. We apply he qadrac formla and hen arrve a R K ϕfx 1, 1 nα + Cnα R α α Cnα K nα3 1 α 1 K ϕn 3/ K 0 + K 1 α. Ths esmae for ϕf s also correc on {x, T 1 dx, x 0, T 1 R} snce 1 T 1. Snce T 1 s arbrary n 0 < T 1 T, we have compleed he proof of Theorem 1.

9 GRADIENT ESTIMATES UNDER RICCI FLOW 171 The local resl above mples a global one. Corollary 1. Le M, g0 be a complee noncompac Remannan manfold who bondary, and sppose g evolves by Rcc flow n sch a way ha K 0 Rc K 1 for [0, T ]. Le be a posve solon o he eqaon x, = 0. Then for x, M 0, T ], we have -15 x, x, α x, x, nα + CK 1 + K 0, for any α > 1, where C depends on n and α only. Proof. By he nform eqvalence of g, we know ha M, g s complee noncompac for [0, T ]. Then le R + n -. Remark 1. When M, g0 s a complee noncompac Remannan manfold, Sh [1989] gves a sffcen condon for he shor me exsence of he Rcc flow: I sffces ha he crvare ensor {R jkl } of g0 sasfes where 0 < κ < + s a consan. R jkl κ on M, Usng Lemma 1, we can also derve a smlar graden esmae on a closed Remannan manfold. Theorem. Le M, g be a closed Remannan manfold, where g evolves by Rcc flow n sch a way ha K 0 Rc K 1 for [0, T ]. If s a posve solon o he eqaon x, = 0, hen for x, M 0, T ], we have -16 x, x, α x, x, nα + nα3 K 0 α 1 + n3/ α K 0 + K 1, for any α > 1, where C depends on n and α only. Proof. Le noaons F and f be as above. Se Fx, = Fx, nα3 K 0 α 1 n3/ α K 0 + K 1. If Fx, nα for any x, M 0, T ], hen he heorem s proved. If -16 doesn hold, hen a he maxmal pon x 0, 0 of Fx,, we have Fx 0, 0 > nα. Nong Fx, 0 = 0, we know here 0 > 0. Then applyng he maxmal prncple, we have a he pon x 0, 0 ha -17 Fx 0, 0 = 0, Fx 0, 0 0, Therefore we oban Fx 0, F F.

10 17 SHIPING LIU Usng Lemma 1 and he rck n calclang -11, we ge nα F 0 F 0 nα4 K 0 α α n K 0 + K 1 Snce F = F + nα3 K α 1 + n3/ α K 0 + K 1 > 0, we ge he neqaly n 0 F F nα nα4 K0 0 0 α α n K 0 + K 1 0. Agan he qadrac formla gves -1 F 0 nα 0 + nα3 K 0 α 1 + n3/ α K 0 + K 1. Ths mples Fx 0, 0 nα, a conradcon. So -16 holds. Remark. In Corollary 1 and Theorem, f K 0 = 0, we can le α 1. α 1 α f F. 0 In fac, he Theorem mproves he graden neqaly n [Genher 00], whch reqres he bondedness of he graden of scalar crvare n addon o he bondedness of he Rcc crvare. Begnnng wh hs resl, we can do hngs smlar o wha was done n [Genher 00], sch as dervng Harnackype neqales. Corollary. Le M, g0 be a complee noncompac Remannan manfold who bondary or a closed Remannan manfold, and sppose g evolves by Rcc flow for [0, T ] n sch a way ha K 0 Rc K 1. If s a posve solon o he eqaon x, = 0, hen for any pons x, 1, y, M 0, T ] sch ha 1 <, we have nε 1 ε γ s - x, 1 y, exp σ ds + C 1 ε K 1 + K 0, for any ε > 1/, where C depends on n and ε only, γs s a smooh crve connecng x and y wh γ1 = x and γ0 = y, and γ s σ s he lengh of he vecor γ s a me σ = 1 s + s 1. Proof. Frs noe ha he graden esmaes n Corollary 1 and Theorem can boh be wren as -3 x, x, α x, x, nα + C n,α K 1 + K 0.

11 GRADIENT ESTIMATES UNDER RICCI FLOW 173 Defne ls = ln γs, 1 s + s 1. I s easy o see ha l0 = ln y, and l1 = ln x, 1. Drec calclaon gves ls γ s = 1 s 1 ε γ s σ CK 1 + K 0 + 4ε n. ε σ s Inegrang hs neqaly over γs, we have ln x, 1 y, = ls s ds ε γ s σ 1 ds + C 1 ε K 1 + K 0 + εn ln / 1. We can also ge a graden esmae for an arbrary solon of he hea eqaon nder Rcc flow on a closed manfold who any crvare condons. The axlary fncon F we ake n he followng proof s nspred by Hamlon s [1995] proof of Sh s [1989] dervave esmaes. Theorem 3. Le M, g be a closed Remannan manfold, where g solves he Rcc flow for [0, T ]. If solves = 0, hen -4 x, 1 max x M x, 0 x, for x, M [0, T ]. Proof. Snce =, we have = Rc, +,. Usng Bochner s formla, hs becomes -5 =. On he oher hand, -6 =. Le F = + A, where A s a consan o be fxed. Then combnng -5 and -6 gves -7 F = + + A F + 1 A. Seng A = 1/ and applyng maxmm prncple, we conclde -8 Fx, max x M Fx, 0 = 1 max x M x, 0. Ths neqaly mples he heorem.

12 174 SHIPING LIU Remark 3. For a posve solon of he hea eqaon on closed manfolds nder Rcc flow, Zhang [006] gves a sronger esmae, 1-9 ln M, where M = max x M x, 0. Smlar o he fac ha he nerpolaon neqaly follows from hs esmae n [Zhang 006], here we ge for any x, y M and 0 < T ha C dx, y, -30 x, y, +, where C = max x M x, The second order graden esmaes Usng Corollary 1, we can generalze o he Rcc flow seng he second order graden esmae for he posve solon of he hea eqaon n [L 1991]. Theorem 4. Le g be a solon o he Rcc flow on a Remannan manfold M n wh n for n some me nerval [0, T ]. Assme ha M, g0 s a complee noncompac manfold who bondary. Sppose he crvare ensor and s frs order covaran dervaves are bonded hrogho by k 1 and k, respecvely. Le be a posve solon o x, = 0. Then for x, M 0, T ], 3-1 x, x, x, + α 5α x, x, x, C k 1 + k /3 + 1, for any α > 1, where C depends on n and α only. To prove he heorem, we se x, Fx, y, = F 1 = x, where β s a consan o be fxed. x, + α x, β x,, x, Lemma. Sppose M, g sasfes he hypoheses of Theorem 4. Then for sffcenly small δ > 0 and γ 1 > 0 and some ε > 0, we have F F log + δα F + δαβ F δα F Ck 1 F Ck 1 4γ 1 Ck4/3 C 4βεn k 1 F/ 4δα 3 δ 1, + α1 δ + k 1

13 GRADIENT ESTIMATES UNDER RICCI FLOW 175 where β = 5α and C depends on n and α. Proof. As n he proof of Lemma 1, choose {x 1, x,..., x n } o be a normal coordnae sysem a a fxed pon. Sbscrps, j and k wll denoe covaran dervaves n he x, x j and x k drecons. We wll calclae he evolon eqaon for F 1. The compaon s a lle long, so we dvde no hree pars. Par 1. We calclae he eqaon for /. I follows from [L 1991] ha jk + j jkk j jk k 3- = 3-3 =. Nong he merc evolves by he Rcc flow, we have = 4 Rc j, j + j j, k j j jk 3 + 3, j = j Ɣ l j l = j + l l Rc jl + j Rc l l Rc j. The Rcc deny gves jkk = kk j + l Rkkl, j l + R kkl l j + R kjl lk + R kjkl l + R jkl,k l + R jkl lk. Combnng he above and sng he Schwarz neqaly, we conclde log Ck 1 Ck. Par. We calclae he eqaon for /. A drec compaon shows α j j j Rc, j j = α + α + α 8α 3 α j j 3 j Rc, + 6α α α + α 4 3 j j j j = α 8α 3 + 6α 4 = α j j j log + α 4α 3 + α j 4.

14 176 SHIPING LIU In he above, he wo Rcc crvare erms generaed by he Rcc deny and he evolon of he merc are canceled. Applyng Yong s neqaly, ha s, j j j 4α 3 1 δα + α 1 δ for any 0 < δ < 1, we conclde α α j log + δα δα 1 δ. Par 3. Usng he formla -4 and Yong s neqaly, we ge for any ε > 0, β β log β j ε βε Rc j. Combnng Pars 1 3 and sng -6, we oban for any 0 < δ < 1 and ε > F 1 F 1 log Ck 1 + δα β j j ε + δα δα 1 δ Ck 4βεn k1. By he defnon of F 1, we have 3-5 F 1 + β, and j = F 1 α 3-6 = F 1 + α + β + β + β F 1 αf 1 αβ Inserng 3-5 and 3-6 no 3-4 and applyng Yong s neqaly o separae he mxed ems, we arrve a 3-7 F 1 F 1 log Ck 1 F Ck 1 /γ δαβ Cβ γ 1 4δα 3 + δ α1 δ δα + β ε + δαf1 j. for any γ 1 > 0. + δαβ F 1 δα F 1 Cδ 1/3 k 4/3 4βεn k 1,

15 GRADIENT ESTIMATES UNDER RICCI FLOW Usng he neqaly we calclae 3-9 γ + γ /, 4δα δαβ Cβ γ 1 1 δαβ γ 4δα 3 + Seng β = 5α, we check ha δ α1 δ δ α1 δ 4δα 3 + Cβ γ 1 δ α1 δ 1 δαβ γ 4δα 3 δ + α1 δ Cβ γ 1 = 8δα γ δ α1 δ γ Cβ γ 1 0 when δ > 0 and γ 1 > 0 are sffcenly small. Then we can ake ε 5/δ sch ha δα β/ε 0. Conseqenly, 3-7 becomes γ F 1 F 1 log Ck 1 F 1 Ck 1 4γ 1 + δαf 1 4δα 3 + δ α1 δ γ + δαβ F 1 / δα F 1 Ck 4/3 4βεn k 1. We complee he lemma by applyng Corollary 1 and nong ha F = F 1 F 1. Proof of Theorem 4. As n he proof of Theorem 1, M, g s complee for [0, T ]. Le ρx, = dx, x 0, and ϕx, = ψρx, /R. Le ϕfx, := ψρx, /RFx,, where x, Q R,T. Sppose x 1, 1 s he pon where ϕf acheves s maxmm n Q R,T1, where 0 < T 1 T.

16 178 SHIPING LIU If x 1, 1 = 0, hen by Corollary 1, we have 3-11 ϕfx 1, 1 = ϕ 1 α β C n,α,β k , whch mples 3-1. Usng he argmens of [Calab 1958] and [L 1991], we can assme ha ϕf s smooh a x 1, 1 and ha ϕfx 1, 1 > 0. As n he proof of Theorem 1, a he pon x 1, 1, we have ϕf C R C k1 F Ck R 1 F + ϕ X F. Applyng Lemma and Yong s neqaly, we oban for s > 0 ha C R C k1 F Ck R 1 F F ϕ sϕ Usng Corollary 1, we have Ck 1 ϕf + δα 1 ϕf Cϕ 1 Fsϕ 4δα 3 δ + α1 δ + δαβϕf / δα ϕf 1/ 1 + k 1 Ck 1 4γ 1 ϕ 1 Ck 4/3 ϕ 1 4βεn ϕ 1 k 1 ϕf/ δαβϕf / δα ϕf Fsϕ δαβ sγ δα γϕf / Cδα + sϕf1/ 1 + K 1. Observe ha δαβ sγ δα γ = 0 when we se s = δα 4/γ 1. Then 3-13 becomes δα ϕf ϕfcs + δα 1/ 1 + k 1 C F 1 s R + C R C k1 F R Ck 1 F Ck 1 ϕf ϕf/ 1 Ck 1 4γ 1 ϕ 1 4βεn ϕ 1 k1 Ck 4/3 ϕ 1 Cϕ 1 4δα 3 + δ α1 δ 1/ 1 + k 1.

17 GRADIENT ESTIMATES UNDER RICCI FLOW 179 Mlplyng hrogh by ϕ 1 and sng 0 ϕ 1, we have δαϕf C ϕf R + C k1 R 1 + 8Cδα /γ1/ 1 + k 1 1 C 1 ϕf δα 4/γ 1R Ck 1 1 Ck 1 4γ 1 1 C 4δα 3 δ + α1 δ 1/ 1 + k 1 1 4βεn k 1 1 Ck4/3 1. Applyng he qadrac formla, we ge 3-17 ϕfx 1, 1 C1 + k k /3 1 + C/R 1. By an argmen smlar o one n he proof of Theorem 1, we conclde 3-18 F 1 x, Ck 1 + k /3 + 1/ + 1/R n Q R,T, where C depends on n and α. Becase M s noncompac, we can le R +. Ths complees he proof of Theorem 4. Acknowledgmen We hank Professor Jay L for hs gdance and encoragemen. We also hank Ynyan Yang and Jn Sn for her very sefl sggesons. References [Calab 1958] E. Calab, An exenson of E. Hopf s maxmm prncple wh an applcaon o Remannan geomery, Dke Mah. J , MR 19,1056e Zbl [Chow and Knopf 004] B. Chow and D. Knopf, The Rcc flow: An nrodcon, Mahemacal Srveys and Monographs 110, Amercan Mahemacal Socey, Provdence, RI, 004. MR 005e: Zbl [Chow e al. 006] B. Chow, P. L, and L. N, Hamlon s Rcc flow, Gradae Sdes n Mahemacs 77, Amercan Mahemacal Socey, Provdence, RI, 006. MR 008a:53068 Zbl [Genher 00] C. M. Genher, The fndamenal solon on manfolds wh me-dependen mercs, J. Geom. Anal. 1:3 00, MR 003a:58034 Zbl [Hamlon 198] R. S. Hamlon, Three-manfolds wh posve Rcc crvare, J. Dfferenal Geom. 17: 198, MR 84a:53050 Zbl [Hamlon 1995] R. S. Hamlon, The formaon of snglares n he Rcc flow, pp n Srveys n dfferenal geomery, II Cambrdge, MA, 1993, eded by S.-T. Ya, Inernaonal, Cambrdge, MA, MR 97e:53075 Zbl [Hamlon 1997] R. S. Hamlon, For-manfolds wh posve soropc crvare, Comm. Anal. Geom. 5:1 1997, 1 9. MR 99e:53049 Zbl [Hamlon 1999] R. S. Hamlon, Non-snglar solons of he Rcc flow on hree-manfolds, Comm. Anal. Geom. 7:4 1999, MR 000g:53034 Zbl

18 180 SHIPING LIU [Kang and Zhang 008] S. Kang and Q. S. Zhang, A graden esmae for all posve solons of he conjgae hea eqaon nder Rcc flow, J. Fnc. Anal. 55:4 008, MR Zbl [L 1991] J. Y. L, Graden esmae for he hea kernel of a complee Remannan manfold and s applcaons, J. Fnc. Anal. 97: 1991, MR 9f:58174 Zbl [L 1994] J. Y. L, The Sobolev neqaly and Sobolev mbeddng heorem for Remannan manfolds wh nonnegave Rcc crvare, Chnese Ann. Mah. Ser. A 15:4 1994, In Chnese. MR 96e:58146 [L and Ya 1986] P. L and S.-T. Ya, On he parabolc kernel of he Schrödnger operaor, Aca Mah. 156: , MR 87f:58156 [Perelman 00] G. Perelman, The enropy formla for he Rcc flow and s geomerc applcaons, preprn, 00. arxv mah.dg/ [Perelman 003] G. Perelman, Rcc flow wh srgery on hree manfolds, preprn, 003. arxv mah.dg/ [Sh 1989] W.-X. Sh, Deformng he merc on complee Remannan manfolds, J. Dfferenal Geom. 30:1 1989, MR 90:580 Zbl [Zhang 006] Q. S. Zhang, Some graden esmaes for he hea eqaon on domans and for an eqaon by Perelman, In. Mah. Res. No. 006, Ar. ID MR 007f:35116 [Zhang 007] Q. S. Zhang, A nform Sobolev neqaly nder Rcc flow, In. Mah. Res. No. IMRN 007, Ar. ID rnm Correced n 007, Ar. ID rnm MR 008g:53083 Receved Sepember 10, 008. Revsed Sepember 8, 008. SHIPING LIU ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE CHINESE ACADEMY OF SCIENCES BEIJING CHINA lshpng@amss.ac.cn