LOWER BOUND OF RICCI FLOW S EXISTENCE TIME

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1 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME GUOYI XU Abstract. Let M n, g) be a compact n-dm n ) manfold wth Rc, and f n 3 we assume that M n, g) R has nonnegatve sotropc curvature. Some lower bound of the Rcc flow s exstence tme on M n, g) s proved. Ths provdes an alternatve proof for the unform lower bound of maxmal exstence tme of a famly of closed Rcc flows, whch appears n Cabezas-Rvas and Wlkng s paper [4]. We can also get an nteror curvature estmates for n = 3 under Rc assumpton among others. Combng these results, we can prove the short tme exstence of the Rcc flow on a large class of 3 dmensonal open manfolds wth Rc, whch especally generalzes the results of Cabezas-Rvas and Wlkng n 3-dmenson. Mathematcs Subject Classfcaton: 35K15, 53C44 1. Introducton In [7], Hamlton ntroduced Rcc flow and proved that the Rcc flow on any compact 3-dmensonal manfold M 3, g) wth Rc > wll deform the metrc g to a metrc wth constant postve sectonal curvature. In [6], Chow proved the above theorem for -dm compact surfaces wth postve curvature. When n 4 and M n, g) R has postve sotropc curvature, n [1], Brendle proved that the Rcc flow deforms M n, g) to a Remannan manfold of constant postve sectonal curvature. From all the above, we know that f Rc > when n = and M n, g) R has postve sotropc curvature when n 3, Rcc flow deforms M n, g) to a Remannan manfold of constant postve sectonal curvature. In all the above case, usng maxmal prncple, t s easy to get the upper bound of exstence tme of the Rcc flow on M n, g). In [13], M. Smon posed the followng queston Problem 3.1 there): Queston 1.1. What elements of the geometry need to be controlled, n order to guarantee that a soluton of Rcc flow does not become sngular? In ths note, we gve a lower bound of exstence tme of the Rcc flow on compact manfold M n, g) wth Rc, and f n 3 we assume that M n, g) R has nonnegatve sotropc curvature. More concretely, we proved the followng: Theorem 1.. Let M n, g) be a compact n-dm n ) manfold wth Rc, and f n 3 we assume that M n, g) R has nonnegatve sotropc curvature. Let x M n be a fxed pont satsfyng Rm r on B x, r ) and B g) x, r ) s Date: October 15, 1. 1

2 GUOYI XU a δ-almost sopermetrcally Eucldean defned n Theorem.1, where r > s some constant and δ = δn), ɛ = ɛ n) are chosen n Remark.. Then there exsts T = Tn, r ) > such that the Rcc flow exsts on M n [, T]. Also on [, T], Rm x, t) r x B t x, 1 ɛ ) r Remark 1.3. One novel thng of such lower bound of exstence tme s that t only depends on the local property of manfolds n a neghborhood of a pont, not globally on the whole manfold. Note for any pont x M n, we can always fnd the correspondng r satsfyng the assumptons n the above theorem, although such r may be very small. Ths also provdes a partal answer to Queston 1.1 when manfolds satsfy the assumptons n the above theorem. Theorem 1.4. Let M n, g) be a compact n-dm manfold wth Rc where n. If n 4, we assume Kg C. Let x M n be a fxed pont satsfyng Rm r on B x, r ) and Vol B x, r )) ν r n for some ν >, r >. Then there exsts T = Tν, r ) > such that the Rcc flow exsts on M n [, T]. Also on [, T], Rm x, t) r x B t x, r ) Furthermore for any r > r, there s a B = Bν, r, r) >, such that on [, T], 1.1) Rmx, t) B + B x B t x, r ) t 4 Remark 1.5. Note that M n, g) R has nonnegatve sotropc curvature s mpled by nonnegatve complex sectonal curvature when n 4 and s equvalent to Rc when n = 3. Theorem 1.4 has stronger assumptons on curvature than Theorem 1.. The stronger assumptons about ntal curvature s to guarantee gettng 1.1), whch s very mportant for our applcaton to short-tme exstence of Rcc flow on noncompact manfolds see the proof of Corollary 3.). From Remark.7, we know the above theorem also holds for Kähler Rcc flow on compact Kähler manfolds wth nonnegatve bsectonal curvature. One motvaton to get such lower bound s for studyng the short-tme exstence of Rcc flow on open manfold M n, g) wth the assumpton n the above theorems. In [4], when M n, g) has nonnegatve complex sectonal curvature, by consderng the doublng of convex sets contaned n a Cheeger-Gromoll convex exhauston of M n, g) and solvng the ntal value problem for the Rcc flow on these closed manfolds, Cabezas-Rvas and Wlkng obtan a sequence of closed solutons of the Rcc flow M, g t)) wth nonnegatve complex sectonal curvature whch subconverge to a soluton of the Rcc flow M n, gt)). Fnally they successfully establshed the short-tme exstence of the Rcc flow on M n, g) wth Kg C among other thngs n [4]. One of the keys to get the above short-tme exstence result s to establsh the followng clam see Proposton 4.3 n [4]): Clam 1.6. There s a lower bound ndependent of ) for the maxmal tmes of exstence T for each Rcc flow M, g t)) obtaned above.

3 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME 3 In [4], to get the above lower bound, the authors used an upper bound of scalar curvature ntegral on unt ball of manfold wth curvature bounded from below, whch was establshed by Petrunn n [1]. Petrunn s beautful result was proved by usng deep results n Alexandrov spaces, although the concluson s n the category of Remannan geometry. One of applcatons of Theorem 1. s to provde a drect proof of Clam 1.6 wthout nvolvng the results n Alexandrov spaces. Moreover usng Theorem 1.4 n 3 dmensonal case, and followng the same strategy n [4], we can get the short-tme exstence of Rcc flow on a larger class of 3-manfolds than the class of 3-manfolds wth nonnegatve sectonal curvature. In [14], M. Smon proved the short-tme exstence of Rcc flow on a class of 3-manfolds wth Rc, other assumptons about curvature near nfnty and njectve radus. It seems that our class of manfolds do not have much overlap wth hs class. A long term goal n 3-dm case s to prove the short-tme exstence of Rcc flow on complete manfolds wth Rc. We hope that our result wll be helpful on studyng ths long term goal. The more ambtous target s to study the short tme exstence of Rcc flow on any complete manfold M n, g), provded that M n, g) R has nonnegatve sotropc curvature. In ths more general scheme, 3-dm case wll be exactly the Rc case. Acknowledgement: The author would lke to thank Bng-Long Chen and Mles Smon for ther nterests and comments. He s also ndebted to Smon Brendle for dscusson, Gang Lu and Yuan Yuan for ther nterests and suggestons. Thanks also go to Zhqn Lu and Jeff Streets.. Perelman s pseudolocalty and local Curvature estmates n Rcc flow In the followng computatons, we wll use some cut-off functons whch are composton of a cut-off functon of R and dstance functon. A cut-off functon ϕ on real lne R, s a smooth nonnegatve nonncreasng functon, t s 1 on, 1] and on [, ). We can further assume that.1) ϕ, ϕ + ϕ ) ϕ 16. Another often used notaton s = t, where s the Laplacan wth the metrc gt). Let us recall Perelman s pseudolocalty theorem frstly. Theorem.1 Perelman s pseudolocalty ). For every α > and n there exsts δ > and ɛ > dependng only on α and n wth the followng property. Let M n, gt)), t [, T] be a smooth soluton of the Rcc flow on compact manfold M n, g)). Let r > and x M n be a pont such that Rx, ) r f or x B g) x, r ) and B g) x, r ) s a δ-almost sopermetrcally Eucldean : Vol Ω) ) n 1 δ)cn Vol Ω) ) n 1

4 4 GUOYI XU for any regular doman Ω B x, r ), where c n n n ω n s the Eucldean sopermetrc constant. Then we have the nteror curvature estmate Rm x, t) α t + 1 ɛ r ) for x B t x, ɛ r ) and t, mn{ ɛ r ), T} ). Remark.. If we choose α = n the above theorem, fx n, x and sutable r, we get that there exsts numercal constants δ = δ, n) and ɛ = ɛ, n) < 1 such that.) Rm x, t) t + 1 ɛ r ) for x B t x, ɛ r ) and t, mn{ ɛ r ), T} ]. We also need a theorem due to Bng-Long Chen n [5]. Theorem.3 Bng-Long Chen). There s a constant C = Cn) wth the followng property. Suppose we have a smooth soluton to the Rcc flow M n, gt)) t [,T] such that B t x, r ), t T, s compactly constaned n M n, and Rm x, ) r f or x B x, r ) Rm x, t) K t where K 1, d t x, t) = dst t x, x) < r and t T. Then we have Rm x, t) e CK r d t x, x) ) whenever t T, d t x, t) = dst t x, x) < r. The followng proposton s motvated by Theorem 3.6 n [5]. The strategy of the proof s applyng Perelman s pseudolocalty theorem to get.), then we can use Theorem.3 to get.3). Proposton.4. Let M n, gt)), t [, T] be a smooth soluton of the Rcc flow on compact manfold M n, g)). Let x M n be a fxed pont satsfyng Rm r on B x, r ) and B g) x, r ) s a δ-almost sopermetrcally Eucldean defned as n Theorem.1, where r > s some constant and δ = δn), ɛ = ɛ n) are chosen as n Remark.. Then there exsts C = Cr, n) > such that.3) Rm r for all x B t x, 1 ɛ r ), t [, mn{t, Cr, n)} ]. Proof: We defne T as the followng: T max { t t T, Rm x, s) r, when x B s x, 1 ɛ ) } r, and s [, t]

5 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME 5 Recall by assumpton, Rmx, ) r on B x, r ). Wthout loss of generalty, we assume that T < mn{t, 1}. Then there s x 1, t 1 ) such that t 1 T, x 1 B t1 x, 1 ɛ ) r, Rm x1, t 1 ) = r. By Theorem.1 and Remark., we get.4) Rmx, t) + T ) ɛ r t Cn, r ) t whenever x B t x, ɛ r ) and t, mn{t, ) ɛ r }]. Now apply Theorem.3 combnng wth.4), we have Rm x, t) Cn, r ) on B t x, 3 4 ɛ ) [ r, where t, mn{t, ) ɛ r } ]. Consder the evoluton equaton of u = ϕ 4d tx,x) ɛ r 1) Rmx, t), where ϕ s the smooth nonnegatve decreasng functon chosen as before. u 4ϕ r d t x, x) 16 ϕ ) Rm ϕ Rm r ϕ Rm + C 1 ϕ Rm 3 where C 1 s some constant dependng only on n. Let u max t) = uxt), t) = max x M 3 ux, t), then by the maxmum prncple and the propertes of ϕ, and usng Lemma 8.3 n [11], at xt), t) we have d + dt u max Cn, r ) Rm + C Rm u max Cn, r ) + Cn, r )u max = C + Cu max where C = Cn, r ) only depends on n, r. Integratng ths nequalty, u max t) [u max ) + 1]e Ct 1 [r 4 + 1]e Ct 1 Hence there exsts Cn, r ) > such that u max t) < 4r 4 when t mn{cn, r ), T }. Then we get Rmx, t) < r on B t x, 1 ɛ ) r when t mn{cν, r ), T }. Recall that there s x 1, t 1 ) such that t 1 T, x 1 B t1 x, 1 ɛ ) r, Rm x1, t 1 ) = r. Therefore T mn{t, Cn, r )}. We recall the defnton of complex sectonal curvature. Defnton.5. Let M n, g) be a Remannan manfold, and consder ts complexfed tangent bundle T C M := T M C. We extend the curvature tensor Rm and the metrc g at p to C-multlnear maps Rm : T C p M) 4 C, g : T C p M) C. The complex sectonal curvature of a -dmensonal complex subspace σ of T C p M s defned by K C σ) = Rmu, v, v, ū) = g Rmu v), u v ), where u and v form any untary bass for σ,.e. gu, ū) = gv, v) = 1 and gu, v) =. We say M n, g) has nonnegatve complex sectonal curvature f K C g. The followng proposton s very close to Proposton.4, and the assumpton about ntal curvature s to guarantee gettng.6), whch s very mportant for our applcaton to Rcc flow on noncompact manfolds.

6 6 GUOYI XU Proposton.6. Let M n, gx, )) be a compact n-dm manfold wth Rc. If n 4, we assume Kg C. Let x M n be a fxed pont satsfyng Rm r on B x, r ) and Vol B x, r )) ν r n for some ν >, r >. Let gx, t), t [, T] be a smooth soluton to the Rcc flow wth gx, ) as ntal metrc. Then there exsts C = Cν, r ) > such that we have.5) Rm r for all x B t x, r ), t mn{t, Cν, r )}. Furthermore for any r > r, there s a B = Bν, r, r) >, such that.6) Rmx, t) B + B t whenever x B t x, r 4 ) and t mn{t, Cν, r )}. Remark.7. Ths proposton generalzes the compact case of Theorem 3.6 n [5] to hgher dmensons. The key observaton s: n the proof of Theorem 3.6 there, we only need two facts. The frst one s Rc durng Rcc flow. The other one s that asymptotc volume ratos AVR) of related ancent solutons wth the same curvature assumptons as ntal metrc are. When we can get these two facts from ntal condton, our proof follows the lnes gven n the proof of Theorem 3.6 n [5], although a number of modfcatons are necessary. For Kähler Rcc flow, we can get smlar results on compact Kähler manfolds wth nonnegatve bsectonal curvature from the related fact AVR = n [9]. Proof: From the assumpton, we get that Rc s preserved from [7] when n =, 3. When n 4, Kg C s preserved from [] also see [1]). Note Kg C mples Rc, hence Rcx, t) on M n [, T]. We defne T as the followng: T max { t t T, Rm x, s) r, when x B sx, r ), and s [, t]} Recall by assumpton, Rmx, ) r on B x, r ). Wthout loss of generalty, we assume that T < T. Then there s x 1, t 1 ) such that t 1 T, x 1 B t1 x, r ), Rm x 1, t 1 ) = r. We wll estmate T from below by a postve constant dependng only on ν, r. We have the followng clam: Clam.8. For any r > r, there s a B = Bν, r, r) >, such that.7) Rmx, t) B + B t whenever x B t x, r 4 ) and t [, T ]. Proof of Clam.8: We argue by contradcton. Suppose.7) does not hold. Then for some fxed r > r, there exsts a sequence of Rcc flow solutons M n, g ) t)), such that there exsts some x, t ), x B g ) t )x, r 4 ) and t [, T ] satsfyng Rmx, t ) g ) B + B t wth lm B =. Usng Lemma.9 n the followng, let A = r 5 B 1 and assume A 1, then we can choose x, t ) wth Q = Rm x, t ) g ) B t 1 such that x B g ) t )x, A B 1 +

7 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME 7 r 4 ) and Rmx, t) g ) t) 4 Q for all d g ) t )x, x) 1 1 A 1 Q 1, t 5n A Q 1 t t. Note t V tb x, r )) = Rx, t)dv t x) Cn)r B x, r V tb x, r ) )) where we used the fact Rcx, t) and the assumpton that Rmx, t) r for all x B t x, r ) and t [, T ] durng the Rcc flow. Hence V g ) t)b x, r )) Cν r n e Cr φν, r ) By B x, r ) B t x, r ) B t x, r), on [, T ], V g ) t)b t x, r)) V g ) t)b x, r )) φν, r ) Scalng metrc by ḡ ) t) = Q g ) t + t Q 1 ) around x, t ). Note d g ) t ) x, x ) r 4 + A B 1 < r 3 If < s 3 r Q 1, note Rcgt)) for t [, T], then usng volume comparson theorem, Vḡ) ) Bḡ) ) x, s ) ) s n = V g) t ) Bg ) t ) V g ) t ) Bg ) t ) x, s Q 1 s Q 1 ) n x, r 3)) )) 3 r) n 1 n φν, r ) r n V g ) t ) Bg ) t )x, 3 r)) 3 r)n Combne Rmx, ) 4, we get n jḡ) ) x ) Cν, r, r) >. Note Rmx, t) 4 for t [ A, ]. Use compactness theorem for Rcc flow, we can extract a subsequence Rcc flow solutons, whch are convergent to a nonflat ancent smooth complete soluton to Rcc flow, whch has max volume growth and Rm 4. When n =, t s obvous that Kg C. We also have Kg C when n = 3, because any 3-dm ancent soluton has Rm by [5]. When n 4, Kg C s preserved. From Lemma 4.5 n [4], we know that for any complete nonflat ancent soluton wth bounded and nonnegatve complex sectonal curvature, ts asymptotc volume rato s. That s contradcton wth the max volume growth property we get above. Therefore Clam.8 s proved. Wthout loss of generalty, we can further assume T 1. Now choose r = 4r and apply Theorem.3, we have Rmx, t) Cν, r ) on B t x, 3 4 r ), t [, T ]. Consder the evoluton equaton of u = ϕ 4d tx,x) r 1) Rmx, t), where ϕ s the smooth nonnegatve decreasng functon chosen as before. u 4ϕ r d t x, x) 16 ϕ ) Rm ϕ Rm r ϕ Rm + C 1 ϕ Rm 3 where C 1 s some constant dependng only on n.

8 8 GUOYI XU Let u max t) = uxt), t) = max x M 3 ux, t), then by the maxmum prncple and the propertes of ϕ, and usng Lemma 8.3 n [11], at xt), t) we have d + dt u max Cr ) Rm + C Rm u max Cν, r ) + Cν, r )u max = C + Cu max where C = Cν, r ) only depends on ν, r. Integratng ths nequalty, u max t) [u max ) + 1]e Ct 1 [r 4 + 1]e Ct 1 Hence there exsts Cν, r ) > such that u max t) < 4r 4 when t mn{cν, r ), T }. Then we get Rmx, t) < r on B t x, r ) when t mn{cν, r ), T }. Recall that there s x 1, t 1 ) such that t 1 T, x 1 B t1 x, r ), Rm x 1, t 1 ) = r. Therefore T mn{t, Cν, r )}. The followng lemma s orgnally due to G. Perelman see secton 1 n [11]). We do some modfcatons to make t sutable for our use n the above proof of Clam.8. Notce that we do NOT have α < 1 1n as n the assumptons of Perelman s Clam 1 and Clam, therefore modfcatons are necessary. Lemma.9. M n, gt)) s a Rcc flow complete soluton on [, T]. B >, r > are some fxed constants, ˆx s some pont n B gˆt) x, r 4 ) for some ˆt, T], and satsfes Rm ˆx, ˆt) gˆt) B +. For any constant A such that < A B, we can Bˆt choose x, t) wth Q = Rm x, t) g t) B t 1 such that x B g t) x, AB 1 + r 4 ) and Rmx, t) gt) 4 Q for all d g t) x, x) 1 1 A Q 1, t 5n A Q 1 t t. Proof: Step I). We frstly prove the followng clam. Clam.1. We can fnd ˇx, ť), Rm ˇx, ť) Bť 1 and ť, T], d gť) ˇx, x ) AB 1 + r 4 such that.8) Rm x, t) 4 Rm ˇx, ť) whenever.9) Rm x, t) Bt 1, < t ť, d gt) x, x ) d gť) ˇx, x ) + A Rm 1 ˇx, ť) For.8) or.9), we say x, t) satsfes.8) or.9) for ˇx, ť). Proof of Clam.1: We use the notaton x 1, t 1 ) for ˆx, ˆt). If we choose ˇx, ť) as x 1, t 1 ), and any x, t) satsfyng.9) wll satsfy.8) for x 1, t 1 ), then we are done. Otherwse, we can fnd x, t ) satsfes.9) but not.8) for x 1, t 1 ). Then.1) Rm x, t ) > 4 Rm x 1, t 1 ) 4B, Rmx, t ) Bt 1 and.11) d gt )x, x ) d gt1 )x 1, x ) + AB 1 r f rac1 + AB 4 By nducton, for any postve nteger k, we can fnd x k, t k ) such that.1) Rm x k, t k ) 4 k 1 B and.13) d gtk )x k, x ) r 4 + AB 1

9 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME 9 Because B gt) x, r 4 + AB 1 ) [, T] s compact, Rm s bounded on t. By.1), the above nducton stops at fnte steps. Hence we fnd ˇx, ť) such that d gť) ˇx, x ) AB 1 + r 4 and Rm ˇx, ť) Bť 1. Further more, f x, t) satsfes.9) for ˇx, ť) then x, t) satsfes.8) for ˇx, ť). The clam s proved. Step II). Let x, t) = ˇx, ť). We wll prove the concluson n the theorem by contradcton. Assume there exsts x, t) such that d g t) x, t) 1 1 A Q 1, t [ t 5n A Q 1, t], but Rm x, t) > 4 Rm x, t). Then Rm x, t) > 4B t 1 > B t 1..14) d g t) x, x ) d g t) x, x ) + d g t) x, x) d g t) x, x ) A Q 1 Hence B g t) x, 1 1 A Q 1 ) B g t) x, d g t) x, x )+ 9 1 A Q 1 ). Suppose that there exsts t 1 t 5n A Q 1 whch s the frst tme, gong backwards n tme from t, such that B g t) x, 1 1 A Q 1 ) ntersects the boundary of B gt1 )x, d g t) x, x ) A Q 1 ). Let x be such at pont of ntersecton, such that.15) d gt1 )x, x ) = d g t) x, x ) A Q 1 By x B g t) x, 1 1 A Q 1 1 ), we get.16) d g t) x, x ) d g t) x, x ) A Q 1 Also we have d gt) x, x ) d g t) x, x ) A Q 1 for t [t 1, t]. Then for t [t 1, t],.17) B gt) x, 1 1 A Q 1 ) B gt) x, d g t) x, x ) + A Q 1 ) By Clam.1, we get Rm x, t) 4 Q n B gt) x, d g t) x, x ) + A Q 1 ) [ t 5n A Q 1, t]. Then Rcx, t) 4n 1) Q for x B gt) x, 1 1 A Q 1 ) B gt) x, 1 1 A Q 1 ), t [t 1, t]. Hence we can apply Lemma 8.3 b) n [11] where we choose K = 4 Q, r = Q 1 ), we have.18) Integrate d dt d gt)x, x ) 1n 1) Q 1.19) d gt1 )x, x ) d g t) x, x ) + 1n 1) Q 1 5n A Q 1 < d g t) x, x ) A Q 1 Ths s the contradcton wth.15). Hence B g t) x, 1 1 A Q 1 ) B gt) x, d g t) x, x )+ 9 1 A Q 1 ), t [ t 5n A Q 1, t]. By x B g t) x, 1 1 A Q 1 ), we get.) d g t) x, x ) d g t) x, x ) A Q 1 By Clam.1, we get Rm x, t) 4 Rm x, t). That s contradcton.

10 1 GUOYI XU 3. Exstence tme estmate of 3-dm Rcc flow Now we are ready to prove our man theorem. Proof of Theorem 1.: If M n, g) s flat, We get our concluson trvally. In the followng of the proof, we assume M n, g) s not flat. We have the followng clam: Clam 3.1. Rmx, t) on M n, gt)) blow up at the same tme durng the Rcc flow. Proof: When n =, t s easy to get from [6]. The clam follows from results n [7] and [8] when n = 3. When n 4, we can get the clam from results n [1] and a general maxmum prncple n [3]. Assume on M n, g), the Rcc flow s maxmal exstence tme nterval s [, T 1 ), where < T 1 <. We defne T as the followng: T max { t t < T 1, Rm x, s) r, when x B t x, 1 ɛ ) } r, and s [, t], From the above clam, we get T < T 1, hence the Rcc flow has the smooth soluton on M n [, T ]. If T < Cr, n), where Cr, n) s from Proposton.4, then there exsts ɛ > such that T + ɛ Cν, r ) and the Rcc flow has smooth soluton on M n [, T + ɛ]. From Proposton.4, we get Rm x, t) r for x B t x, 1 ɛ ) r and t [, T + ɛ]. It s contradcton wth the defnton of T. Hence T Cr, n), we get our concluson. Proof of Theorem 1.4: Recall that M n, gx, )) R has nonnegatve sotropc curvature s mpled by nonnegatve complex sectonal curvature when n 4 and s equvalent to Rc when n = 3. Hence we stll have Clam 3.1 under the assumpton of Theorem 1.4. The rest proof of Theorem 1.4 s smlar to the above proof except that we use Proposton.6 nstead of Proposton.4. Corollary 3.. Let M 3, g) be a complete noncompact 3-dmensonal manfold wth Rc. Assume there exsts an exhauston {Ω } =1 of M3, g) wth the property Ω Ω +1 for eahc, and for each Ω there exsts M, g ) ) wth Rcg ) ) such that Ω can be sometrcally mbedded nto M, g ) ). Then there exsts a constant T > such that the Rcc flow has a complete soluton on M 3 [, T] wth g) = g. Proof: The proof of the corollary s smlar to the proof of Theorem 4.7 n [4], except that we use Theorem 1.4 here nstead of Proposton 4.3 and 4.6 there. Note n ths case, the estmate 1.1) s smlar to Proposton 4.6 there. Example 3.3. From Proposton 6.7 n [4], f M 3, g) s a 3-dm open manfold wth Rm, then there exsts an exhauston {Ω } =1 of M3, g) wth the property Ω Ω +1 for eahc. And for each Ω, there exsts M, g ) ) wth Rmg ) ) such that Ω can be sometrcally mbedded nto M, g ) ). By the Corollary above, the Rcc flow has a complete soluton on M 3 [, T] for some T >, ths s the 3-dm case of Theorem 1 n [4]. References [1] Smon, Brendle: A general convergence result for the Rcc flow n hgher dmensons. Duke Math. J ), no. 3,

11 LOWER BOUND OF RICCI FLOW S EXISTENCE TIME 11 [] Smon, Brendle & Rchard, Schoen: Manfolds wth 1/4-pnched curvature are space forms. J. Amer. Math. Soc. 9), no. 1, [3] Smon, Brendle & Rchard, Schoen: Classfcaton of manfolds wth weakly 1/4-pnched curvatures. Acta Math. 8), no. 1, [4] Esther, Cabezas-Rvas & Burkhard, Wlkng: How to produce a Rcc Flow va Cheeger- Gromoll exhauston. arxv:math.dg/ [5] Bng-Long Chen: Strong unqueness of the Rcc flow. J. Dfferental Geom. 8 9), no., [6] Bennett, Chow: The Rcc flow on the -sphere. J. Dfferental Geom ), no., [7] Rchard S. Hamlton: Three-manfolds wth postve Rcc curvature. J. Dfferental Geom ), no., [8] Rchard S. Hamlton: Four-manfolds wth postve curvature operator. J. Dfferental Geom ), no., [9] Le, N: Ancent solutons to Khler-Rcc flow. Math. Res. Lett. 1 5), no. 5-6, [1] Le, N & Jon, Wolfson: Postve Complex Sectonal Curvature, Rcc Flow and the Dfferental Sphere Theorem. arxv:math.dg/76.33 [11] G. Perelman: The entropy formula for the Rcc flow and ts geometrc applcatons. arxv:math.dg/11159 [1] A. Petrunn: An upper bound for the curvature ntegral, Algebra Analz 8), no., Russan); translaton n St. Petersburg Math. J. 9), no., [13] Mles, Smon: Rcc flow of almost non-negatvely curved three manfolds. J. Rene Angew. Math. 63 9), [14] Mles, Smon: Rcc flow of non-collapsed three manfolds whose Rcc curvature s bounded from below. J. Rene Angew. Math. 66 1), Mathematcs Department, Unversty of Calforna, Irvne, CA 9697 E-mal address: guoyxu@math.uc.edu