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ASIAN J. MATH. c 016 International Press Vol. 0, No. 5, pp. 903 918, Noveber 016 004 THE NORMALIZED RICCI FLOW ON FOUR-MANIFOLDS AND EOTIC SMOOTH STRUCTURES MASASHI ISHIDA Abstract. We shall prove that, for every natural nuber l there exists a closed topological 4- anifold l which adits sooth structures for which non-singular solutions of the noralized Ricci flow exist, but also adits sooth structures for which no non-singular solution of the noralized Ricci flow exists. Hence, in diension 4, sooth structures becoe definite obstructions to the existence of non-singular solutions to the noralized Ricci flow. Key words. Ricci flow, non-singular solution, exotic sooth structure. AMS subject classifications. 57R57, 53C1, 53C44. 1. Introduction. Let be a closed oriented Rieannian anifold of diension n. The noralized Ricci flow [14] on is the following evolution equation: ( s gdμ g t g = Ric g + n dμ g ) g, (1.1) where Ric g and s g denote respectively the Ricci curvature and the scalar curvature of the evolving Rieannian etric g, vol g := dμ g and dμ g is the volue easure with respect to g. Recall that a solution {} to the noralized Ricci flow on a tie interval [0,T) is said to be axial if it cannot be extended past tie T. A axial solution{},t [0,T), to the noralized Ricci flow on is called non-singular [15] if T = and the Rieannian curvature tensor R of satisfies sup R <. [0,T ) As a pioneer work, Hailton [14] proved that there exists a unique non-singular solution to the noralized Ricci flow on 3-anifolds if the initial etric is positive Ricci curvature. Moreover, Hailton [15] classified non-singular solutions to the noralized Ricci flow on 3-anifolds. These fundaental works were very iportant for understanding long-tie behavior of solutions of the Ricci flow on 3-anifolds. On the other hand, though any authors study the properties of non-singular solutions in higher diensions, the existence and non-existence of non-singular solutions to the noralized Ricci flow are still ysterious in general. The ain purpose of the current article is to study, fro the gauge theoretical point of view, this proble in case of diension four. We shall point out that the difference between existence and non-existence of non-singular solutions to the noralized Ricci flow strictly depend on one s choice of sooth structure. The ain result is Theore 1.3 stated below. In an iportant work [10], Fang, Zhang and Zhang studied the properties of nonsingular solutions in higher diensions. Let be a closed oriented sooth 4- anifold and suppose that there is a non-singular solution to the noralized Ricci flow on. Then, one of fundaental discoveries due to [10] is that if the solution satisfies ŝ c < 0, where the constant c is independent of t and define as Received July 4, 014; accepted for publication May 6, 015. Matheatical Institute, Tohoku University, Sendai, 980-8578, Japan (asashi.ishida@.tohoku. ac.jp). 903

904 M. ISHIDA ŝ g := in x s g (x) for a given Rieannian etric g, the 4-anifold ust satisfy the following topological constraint on the Euler characteritic χ() and signature τ() of: χ() 3 τ(). In this article, we shall call this Fang-Zhang-Zhang inequality (or, for brevity, FZZ inequality) we shall also call χ() > 3 τ() the strict FZZ inequality. The FZZ inequality gives us, under the above condition on ŝ, the only known topological obstruction to the existence of non-singular solutions to the noralized Ricci flow. It is also known that any Einstein 4-anifold ust satisfy the sae bound χ() 3 τ() which is so called Hitchin-Thorpe inequality [40, 16]. Since any Einstein etric is a non-singular solution, FZZ inequality can be seen as a generalization of Hitchin-Thorpe inequality to Ricci flow case. On the other hand, there is a natural diffeoorphis invariant arising fro a variational proble for the total scalar curvature of Rieannain etrics. As was conjectured by Yaabe [43], and later proved by Trudinger, Aubin, and Schoen [, 30, 37, 41], every conforal class on any sooth copact anifold contains a Rieannian etric of constant scalar curvature. For each conforal class [g] ={vg v : R + }, we are able to consider an associated nuber Y [g] which is so called Yaabe constant of the conforal class [g]. The Trudinger-Aubin-Schoen theore tells us that this nuber is actually realized as the constant scalar curvature of soe unit volue etric in the conforal class [g]. Then, Kobayashi [1] and Schoen [38] independently introduced the following invariant which is so called Yaabe invariant of : Y() =supy [g], C where C is the set of all conforal classes on. It is known that Y() 0ifand only if does not adit a etric of positive scalar curvature. It is also known that the Yaabe invariant is sensitive to the choice of sooth structure of a 4-anifold. After the celebrated works of Donaldson [9] and Freedan [11], it now turns out that any exotic sooth structures exist in diension 4. We are able to construct any exaples of topological 4-anifolds aditting distinct sooth structures for which values of the Yaabe invariants are different by using the result [18] of LeBrun with the author of the current article. We shall observe that the above condition ŝ c<0 is closely related to the negativity of the Yaabe invariant. More precisely, in Proposition. proved in Section, we shall see that the condition ŝ c<0isalways satisfied for any solution to the noralized Ricci flow if a given sooth Rieannian anifold of diension n 3hasY() < 0. By results proved by [1] and [10], we are also able to see that, if a copact topological 4-anifold M adits a sooth structure Z with Y < 0 and for which there exists a non-singular solution to the noralized Ricci flow, then the strict FZZ inequality χ(z) > 3 τ(z) ust hold, where we denote the copact topological 4-anifold M adits the sooth structure Z by Z. Let us here ephasize that χ(z) > 3 τ(z) is just a topological constraint, is not a differential topological one. This observation and the special feature of sooth structures in diension 4 naturally lead us to ask the following: Question 1.. Let be any copact topological 4-anifold which adits at least two distinct sooth structures Z i with negative Yaabe invariant Y < 0.

RICCI FLOW AND SMOOTH STRUCTURES 905 Suppose that, for at least one of these sooth structures Z i, there exist non-singular solutions to the the noralized Ricci flow. Then, for every other sooth structure Z i with Y < 0, are there always non-singular solutions to the noralized Ricci flow? Since adits, for at least one of these sooth structures Z i, non-singular solutions to the the noralized Ricci flow, we are able to conclude that χ(z i ) > 3 τ(z i ) holds for every i. Notice that this is equivalent to χ() > 3 τ(). Hence, even if there are always non-singular solutions to the noralized Ricci flow for every other sooth structure Z i, it dose not contradict the strict FZZ inequality. Interestingly, the ain result of the current article tells us that the answer to Question 1. is negative as follows: Theore 1.3. For every natural nuber l, there exist a siply connected copact topological non-spin 4-anifold l satisfying the following properties: (1) l adits at least l different sooth structures Ml i with Y < 0 and for which there exist non-singular solutions to the the noralized Ricci flow. Moreover the existence of the solutions forces the strict FZZ inequality χ >3 τ as a topological constraint, () l also adits infinitely any different sooth structures N j l with Y < 0 and for which there exists no non-singular solution to the noralized Ricci flow. Notice that Freedan s classification [11] iplies that l above ust be hoeoorphic to a connected su pcp #qcp,wherecp is the coplex projective plane and CP is the coplex projective plane with the reversed orientation, and p and q are soe appropriate positive integers which depend on the natural nuber l. Notice also that, for the standard sooth structure on pcp #qcp,wehavey > 0 because, by a result of Schoen and Yau [36] or Groov and Lawson [13], there exists a Rieannian etric of positive scalar curvature for such a sooth structure. Hence, sooth structures which appear in Theore 1.3 are far fro the standard sooth structure. To the best of our knowledge, Theore 1.3 is the first result which shows that, in diension four, sooth structures becoe definite obstructions to the existence of non-singular solutions to the noralized Ricci flow. Naely, Theore 1.3 teaches us that the existence or non-existence of non-singular solutions depends strictly on the diffeotype of a 4-anifold and it is not deterined by hoeotype alone. This gives a new insight into the property of solutions to the Ricci flow on 4-anifolds. To prove the non-existence result in Theore 1.3, we need to prove new obstructions to the existence of non-singular solutions to the noralized Ricci flow. Indeed, it is the ain non-trivial step in the proof of Theore 1.3. For instance, we shall prove that, for any closed syplectic 4-anifold with b + () andχ()+3τ() > 0, where b + () stands for the diension of a axial positive definite subspace of H (, R) with respect to the intersection for, there is no non-singular solution of the noralized Ricci flow on a connected su M := #kcp if 3k >χ()+3τ() holds. See Corollary.15 stated below. See also Theores.1 and.16 below for ore general obstructions. We shall use the Seiberg-Witten onopole equations [4] to prove the obstructions. We should notice that, under the sae condition with the above, LeBrun [7] firstly proved that the above 4-anifold M cannot adit any Einstein etric by using Seiberg-Witten onopole equations [4]. However, notice that non-singular solutions do not necessarily converge to sooth Einstein etrics. See also [10]. Hence, non-existence result on non-singular solutions proved in the current article never follow fro the obstruction of LeBrun in general. In this sense,

906 M. ISHIDA the obstructions proved in Section are new. Acknowledgeents. The current article is a refined version of original results announced in the archive [19]. The article [19] was written when the author was visiting State University of New York at Stony Brook in 006. I would like to express y deep gratitude to Claude LeBrun for his encourageent and hospitality. I would like to thank the Departent of Matheatics of SUNY at Stony Brook for their hospitality and nice atosphere during the preparation of the article. The author is supported in part by Grant-in-Aid for Scientific Research (C) 5400074.. Ricci flow solutions with bounded scalar curvature. Let be a closed oriented Rieannian anifold of diension n 3, and [g] ={ug u : R + } is the conforal class of an arbitrary etric g. Trudinger, Aubin, and Schoen [, 30, 37, 41] proved every conforal class on any sooth copact anifold contains a Rieannian etric of constant scalar curvature. Such a etric ĝ can be constructed by iniizing the Einstein-Hilbert functional ĝ s ĝ dμĝ ( dμ ĝ ) n n dμ ĝ ) n n aong all etrics conforal to g. Notice that, by setting ĝ = u 4/(n ) g,wehave [ ] s ĝ dμĝ s g u +4 n 1 n u dμ g ( = ) (n )/n. ( un/(n ) dμ g As was already entioned in Introduction, associated to each conforal class [g], we are able to define the following nuber which is called Yaabe constant of [g]: Y [g] = inf s ĝ dμĝ (. ĝ [g] Equivalently, Y [g] = inf u C + () dμ ĝ ) n n [ ] s g u +4 n 1 n u dμ g ) (n )/n, ( un/(n ) dμ g where C + () is the set of all positive functions u : R +. Then the Yaabe invariant [1, 38] of is defined by Y() =sup [g] C Y [g],wherec is the set of all conforal classes on. It is also known [6] that the following holds if Y() 0: ( Y() = inf g s g n dμg ) n, (.1) where supreu is taken over all sooth etrics g on. Proposition.. Let be a closed oriented Rieannian anifold of diension n 3 and with Y() < 0. If there is a solution {}, t [0,T), to the noralized Ricci flow, then the solution satisfies ŝ := in s Y() (x) < 0. x (vol g(0) ) /n

RICCI FLOW AND SMOOTH STRUCTURES 907 Proof. Suppose that there is a solution {}, t [0,T) to the noralized Ricci flow. Let us consider the Yaabe constant Y [] of a conforal class [] of a etric for any t [0,T). By definition, we have We therefore obtain Y() Y [] = Y() inf u C + () ŝ ( inf u C + () inf u C + () [ ] s u +4 n 1 n u dμ ) (n )/n. ( un/(n ) dμ ( in s u +4 n 1 x n u ) dμ ( ) n /n un/(n ) dμ u dμ ( un/(n ) dμ ) n /n ), where notice that ŝ g := in x s g (x). If ŝ 0 holds, then the above estiate tells us that Y() 0. Since we assue that Y() < 0, we are able to conclude that ŝ < 0 ust hold. On the other hand, the Hölder inequality tells us that the following inequality holds: This iplies that ( u dμ ( inf u C + () = n /n( ) /n u n/n dμ ) dμ ) n /n(vol u n/n dμ ) /n. u dμ ( un/n dμ ) n /n (vol ) /n. Since we have ŝ < 0, this also iplies ( ŝ inf u dμ ) ( ) n /n ŝ (vol ) /n. un/n dμ u C + () We therefore obtain Y() ŝ ( inf u C + () ŝ (vol ) /n. u dμ ) ( ) n /n un/n dμ On the other hand, notice that the noralized Ricci flow preserves the volue of the solution. Therefore, we have vol = vol g(0) for ant t [0,T). Hence, we get the desired bound for any t [0,T): ŝ Y() (vol ) = Y() < 0. /n (vol g(0) ) /n

908 M. ISHIDA To prove the ain result of the current article, we shall use the Seiberg-Witten theory [4, 33]. For the convenience of the reader who is unfailiar with Seiberg- Witten theory, we shall recall briefly the definition of the Seiberg-Witten onopole equations. Let be a closed oriented Rieannian 4-anifold with b + (). Recall that a spin c -structure Γ on a sooth Rieannian 4-anifold induces a pair of spinor bundles S ± Γ which are Heritian vector bundles of rank over. A Rieannian etric on and a unitary connection A on the deterinant line bundle L Γ := det(s + Γ ) induce the twisted Dirac operator D A :Γ(S + Γ ) Γ(S Γ ). The Seiberg-Witten onopole equations over are the following syste for a unitary connection A A LΓ and a spinor φ Γ(S + Γ ): D A φ =0,F + A = iq(φ), (.3) where F + A is the self-dual part of the curvature of A and q : S+ Γ + is a certain natural real-quadratic ap satisfying q(φ) =(1/ ) φ,where + is the bundle of self-dual -fors. We recall the definition of onopole class [, 7, 17, 18, 9]. Definition.4. Let be a closed oriented sooth 4-anifold with b + (). An eleent a H (, Z)/torsion H (, R) is called onopole class of if there exists a spin c -structure Γ with c R 1 (L Γ ) = a which has the property that the corresponding Seiberg-Witten onopole equations (.3) have a solution for every Rieannian etric on. Herec R 1 (L Γ ) is the iage of the first Chern class c 1 (L Γ ) of the coplex line bundle L Γ in H (, R). There are several ways to detect the existence of onopole classes. For exaple, if is a closed syplectic 4-anifold with b + (), then c 1 () is onopole class by the celebrated result of Taubes [39], where c 1 () is the first Chern class of the canonical bundle of. This is proved by thinking the oduli space of solutions of (.3) as a cycle which represents an eleent of the hoology of a certain configuration space. For any closed oriented sooth 4-anifold with b + (), one can define the integer valued Seiberg-Witten invariant SW (Γ ) Z for any spin c -structure Γ by integrating a cohoology class on the oduli space of solutions of (.3) associated with Γ : SW : Spin() Z, where Spin() isthesetofallspin c -structures on. Taubes [39] proved that SW (ˆΓ ) 1 (od ) holds for any closed syplectic 4-anifold with b + (), where ˆΓ is the canonical spin c -structure induced fro the syplectic structure. This iplies that c 1 () is a onopole class of. There is a sophisticated refineent of the idea of this construction. It detects the presence of a onopole class by an eleent of a stable cohootopy group. This is due to Bauer and Furuta [3, 4]. They interpreted (.3) as a ap between two Hilbert bundles over the Picard tours of a 4-anifold. The ap is called the Seiberg-Witten ap. Roughly speaking, the cohootopy refineent of the Seiberg-Witten invariant is defined by taking an stable cohootopy class of the finite diensional approxiation of the Seiberg-Witten ap. The invariant takes its value in a certain coplicated stable cohootopy group. We shall call the invariant stable cohootopy Seiberg-Witten invariant. The nontriviality of this invariant also iplies the existence of onopole classes. For a Rieannian etric g on, the cohoology group H (; R) is identified

RICCI FLOW AND SMOOTH STRUCTURES 909 with the space of g-haronic -fors, and is decoposed into the direct su of the g-self-dual part H g + and the g-anti-self-dual part Hg. For a cohoology class a H (; R)/ Tor H (; R), let a + H g + be the g-self-dual part of a. We shall use the following curvature bounds proved by LeBrun [7, 9]: Proposition.5. Let (, g) be a closed oriented Rieannian 4-anifold. If a is a onopole class of, then s gdμ g 3π (a + ), (.6) (s g 6 W + g ) dμ g 7π (a + ) (.7) hold for every Rieannian etric g on, wheres g is the scalar curvature and W + is the self-dual part of the Weyl curvature. By using (.6), we obtain a result on upper bounds on Yaabe invariants: Proposition.8. Let N beaclosedorientedsooth4-anifoldwithb + (N) =0. Let be a closed alost-coplex 4-anifold with b + () and c 1() =χ()+ 3τ() 0. Assue that SW (Γ ) 0 holds, where Γ is the spin c -structure copatible with the alost-coplex structure. Then the Yaabe invariant of M = #N satisfies Y(M) 4π c 1 () 0. (.9) Proof. It is known [3, 5] that there is a coparison ap between the stable cohootopy refineent of Seiberg-Witten invariant and the integer valued Seiberg- Witten invariant. In particular, Proposition 5.4 in [5] tells us that the coparison ap becoes isoorphis when the given 4-anifold is alost-coplex and b + > 1. Hence, the value of the stable cohootopy Seiberg-Witten invariant [3, 4] of for the spin c -structure Γ copatible with the alost coplex structure is non-trivial if SW (Γ ) 0 holds. Then, the proofs of Proposition 6 and Corollary 8 in [18] (see also Theore 8.8 in [5]) iply that M has non-trivial stable cohootopy Seiberg- Witten invariants and a := ±c 1 ()+ k ±E i is a onopole class of #N, where E 1,E,,E k be a set of generators for H (N,Z)/torsion relative to which the intersection for is diagonal, c 1 () isthe first Chern class of the canonical bundle of the alost-coplex 4-anifold, the ± signs are arbitrary and are independent of one another. Then, by the standard arguent (for instance, see Corollary 11 of [18]), we are able to prove (a + ) (c + 1 ()). Since (c + 1 ()) c 1() holds, we obtain (a + ) c 1(). (.10) By this and (.6), we get inf s gdμ g 3π c 1(). (.11) g M

910 M. ISHIDA On the other hand, notice that the non-triviality of stable cohootopy Seiberg-Witten invariants of M forces that M cannot adit any etric of positive scalar curvature. This iplies that Y(M) 0. Therefore, we are able to obtain the forula (.1) in the case of n = 4. Hence the desired result now follows fro (.11). The ain result of this section is the following: Theore.1. Let N beaclosedorientedsooth4-anifoldwithb + (N) =0. Let be a closed alost-coplex 4-anifold with b + () and c 1() =χ()+ 3τ() > 0. Assue that SW (Γ ) 0holds, where Γ is the spin c -structure copatible with the alost-coplex structure. Suppose that there is a long-tie solution {} of (1.1) on M := #N with scalar curvature s g < C for a constant C independent of t [0, + ). Then the following holds: 4b 1 (N)+b (N) 1 3 c 1 (). Proof. Suppose that there is a long-tie solution {} of (1.1) on M with scalar curvature s g < C for a constant C independent of t. Since we have Y(M) 4π c 1 () < 0 by (.9), where we also used c 1() =χ()+3τ() > 0, Proposition. iplies 4π ŝ c (vol g(0) ) 1/ 1 () < 0. Hence {} with s g <C satisfies ŝ D<0 by setting 4π D = c (vol g(0) ) 1/ 1 (). Notice that D is independent of t. Therefore, the following holds by Lea 3.1 of [10]: r dμ dt <. 0 M This iplies that +1 r dμ dt 0 (.13) M holds when +. On the other hand, we also get the following by the Chern- Gauss-Bonnet forula and the Hirzebruch signature forula, (χ +3τ)(M) = 1 ( s 4π M 4 + W + ) r dμ g (t). Therefore, we have (χ +3τ)(M) = +1 = 1 4π ( ) (χ +3τ)(M) dt +1 M ( s 4 + W + ) r dμ g (t)dt.

RICCI FLOW AND SMOOTH STRUCTURES 911 By taking + and using (.13), we get 1 +1 ( ) s (χ +3τ)(M) = li 4π 4 + W + dμ g (t)dt. (.14) On the other hand, the Cauchy-Schwarz inequality and the triangle inequality iply that the following (see [7]): ( ) s 4 + W + dμ 1 ( s dμ 6 W ) +. 7 M This inequality, (.7) and (.10) iply ( ) 1 s 4π M 4 + W + dμ 3 (χ +3τ)() = 3 c 1(). Therefore, we have 1 +1 4π Hence we get li M 1 4π M M ( ) s 4 + W + dμ dt +1 M This inequality and (.14) iply +1 3 c 1()dt = 3 c 1(). ( ) s 4 + W + dμ dt 3 c 1(). (χ +3τ)(M) 3 c 1 (). We also have χ(m)+3τ(m) =c 1 () (4b 1(N)+b (N)),whereweusedb + (N) =0. Therefore, the following holds: c 1 () (4b 1(N)+b (N)) 3 c 1 (). This iplies the desired result. Theore.1 and the celebrated result of Taubes [39] iply Corollary.15. Let beaclosedsyplectic4-anifoldwithb + () and c 1() =χ() +3τ() > 0. Suppose that there is a long-tie solution {} of (1.1) on M := #kcp with scalar curvature s g <Cfor a constant C independent of t [0, + ). Then k 1 3 c 1 (). We are also able to prove the following result: Theore.16. For i =1,, 3, 4, let i be a closed alost-coplex 4-anifold whose integer valued Seiberg-Witten invariant satisfies SW i (Γ i ) 1 (od ),

91 M. ISHIDA where Γ i is the spin c -structure copatible with the alost-coplex structure. Assue that the following conditions are satisfied: b 1 ( i )=0, b + ( i ) 3(od4), 4 b + ( i ) 4(od8), (.17) c 1 ( i)= (χ( i )+3τ( i )) > 0, (.18) where j =, 3, 4. Let N beaclosedorientedsooth4-anifoldwithb ( + (N)) =0. Suppose that there is a long-tie solution {} of (1.1) on M j := # j i #N with scalar curvature s g <Cfor a constant C independent of t [0, + ), where j =, 3, 4. Then 4(j 1) + 4b 1 (N)+b (N) 1 3 c 1 ( i). Proof. By Corollary 11 of [18], there is a onopole class a of M j such that (a + ) c 1( i ). This and (.6) tell us that Yaabe invariant of M j satisfies Y(M j ) 4π c 1 ( i) < 0, where we used (.18). For ore details, see [18]. Suppose that there is a longtie solution {} of (1.1) on M j with s g <Cfor a constant C independent of t [0, + ). By Proposition., we are able to get ŝ E<0, where 4π E = c (vol g(0) ) 1/ 1 ( i). Then Lea 3.1 of [10] tells us that 0 M j r dμ dt <. This iplies that +1 M j r dμ dt 0 (.19)

RICCI FLOW AND SMOOTH STRUCTURES 913 holds when +. Therefore, we have 1 +1 (χ +3τ)(M j ) = li 4π M j ( ) s 4 + W + dμ g (t)dt. (.0) As in the proof of Theore.1, we are also able to get ( ) 1 s 4π M 4 + W + dμ c 1 3 ( i). Therefore, we have li 1 4π This and (.0) iply +1 M ( ) s 4 + W + dμ dt 3 (χ +3τ)(M j ) 3 On the other hand, a direct coputation iplies (χ +3τ)(M j )= c 1 ( i). c 1( i ). (χ( i )+3τ( i )) + (χ(n)+3τ(n)) 4j = (4b 1 (N)+b (N)) 4(j 1) + whereweusedb + (N) = 0. Therefore, we get (4b 1 (N)+b (N)) 4(j 1) + c 1 ( i) 3 c 1( i ), c 1 ( i). This iplies the desired result as proised. 3. Proof of Theore 1.3. In what follows, we shall use the following notations for any 4-anifold : χ h () := 1 ( ) χ()+τ(), c 1 () :=χ()+3τ(). 4 We shall prove the following result: Proposition 3.1. For every δ>0, there exists a constant d δ > 0 satisfying the following property: every lattice point (α, β) satisfying 0 <β<(6 δ)α d δ (3.) is realized by (χ h,c 1) of infinitely any pairwise non-diffeoorphic siply connected syplectic 4-anifolds with the following properties: (1) each syplectic 4-anifold N is non-spin,

914 M. ISHIDA () N has Y(N) < 0, (3) there exists no non-singular solution of the noralized Ricci flow on N. Proof. Building upon syplectic su construction due to Gopf [1] and gluing forula of Seinerg-Witten invariants due to Morgan-Mrowka-Szabó [31] and Morgan- Szabó-Taubes [3], a nice result on infinitely any pairwise non-diffeoorphic siply connected syplectic 4-anifolds is proved in [6]. In particular, infinitely any sooth structures are given by perforing the logarithic transforation in the sense of Kodaira. Theore 4 of [6] tells us that, for every δ>0, there exists a constant d δ > 0 satisfying the following property: every lattice point (α, β) satisfying 0 <β (9 δ)α d δ is realized by (χ h,c 1) of infinitely any pairwise non-diffeoorphic siply connected syplectic 4-anifolds. In particular, each syplectic 4-anifold satisfies c 1 () = β>0 and we are able to see that b + () by the construction. By Corollary.15, we conclude that, if a positive integer k satisfies k> 1 3 c 1 () =β 3, then there exists no non-singular solution to the noralized Ricci flow on the syplectic 4-anifold N := #kcp. Moreover, the syplectic 4-anifold N is non-spin. These non-spin syplectic 4-anifolds N actually cover the area (3.), where notice that χ h (N) =χ h (), c 1 (N) =β k< 3 β. Moreover, under the connected su with CP, the infinitely any different sooth structures reain distinct as was already noticed in [6]. Finally, since has nontrivial valued Seiberg-Witten invariants by a result of Taubes [39], the bound (.9) tells us that Y(N) 4π c 1 () = 4π β<0. Therefore, we get the desired result. We also have Proposition 3.3. For every positive integer l>0, there are l-tuples of siply connected spin and non-spin algebraic surfaces with the following properties: (1) these are hoeoorhic, but are pairwise non-diffeoorphic, () for every fixed l>0, the ratios c 1 /χ h of the l-tuples are dense in the interval J := [4, 8], (3) each algebraic surface M has Y(M) < 0, (4) there exists a non-singular solution to the the noralized Ricci flow on M. Moreover the existence of the solution forces the strict FZZ inequality χ(m) > 3 τ(m). Proof. Salvetti [35] proved that, for any k>0, there exists a pair (χ h,c 1 )such that for this pair one has at least k hoeoorphic algebraic surfaces with different divisibilities for their canonical classes by taking iterated branched covers of the projective plane. This construction is fairly generalized in [6]. By Corollary 1 of [6],

RICCI FLOW AND SMOOTH STRUCTURES 915 we know that, for every l, therearel-tuples of siply connected spin and non-spin algebraic surfaces with aple canonical bundles which are hoeoorphic, but are pairwise non-diffeoorphic. Moreover, it was also shown that, for every fixed l, the ratios c 1 /χ h of the l-tuples are dense in the interval J. Therefore, to prove this proposition, it is enough to prove (3) and (4). We notice that one can see that each such an algebraic surface M has b + (M) 3 by the construction. Now, the negativity of the Yaabe invariant of the algebraic surface M is a direct consequence of Proposition.8. In fact, the canonical bundle of each algebraic surface M is aple and hence c 1 (M) < 0. In particular, since M is a Kähler surface with b + (M) 3 and c 1 (M) > 0, Proposition.8 tells us that Y(M) = 4π c 1 (M) < 0. Hence we proved (3). Moreover, (4) follows fro the celebrated result of Cao [7, 8] because each algebraic surface M has aple canonical bundle and hence c 1 (M) < 0. We therefore conclude that, for the initial etric g 0 which is chosen to represent the first Chern class, there always exists a non-singular solution to the noralized Ricci flow and it converges to an Einstein etric of negative scalar curvature as t. On the other hand, it was proved in [10] that the existence of non-singular solutions on 4-anifold with λ() < 0 iplies χ() 3 τ() 1 96π λ (), where λ() istheperelanλ-invariant [34, 0] of. It is also known [1] that Y() =λ() holdsify() 0. Therefore, we are able to conclude that M ust satisfy the strict FZZ inequality χ(m) > 3 τ(m) becausewehavey(m) < 0. Propositions 3.1 and 3.3 enable us to prove the ain result as follows: Proof of Theore 1.3. Proposition 3.3 tells us that, for every positive integer l>0, we are always able to find l-tuples Ml i of siply connected non-spin algebraic surfaces of general type and these are hoeoorhic, but are pairwise nondiffeoorphic. And the ratios c 1 /χ h of Ml i are dense in the interval J := [4, 8] for every fixed l>0. Moreover, Proposition 3.3 tells us that Y(Ml i ) < 0 holds and, on each of Ml i, there exists a non-singular solution of the the noralized Ricci flow and the existence of the solution forces the strict FZZ inequality χ >3 τ. On the other hand, Proposition 3.1 tells us that any pair (α, β) in the area (3.) can be realized by (χ h,c 1) of infinitely any pairwise non-diffeoorphic siply connected non-spin syplectic 4-anifolds with Y < 0 and on each of which there exists no non-singular solution of the noralized Ricci flow. Notice that the ratios c 1 /χ h of these non-spin syplectic 4-anifolds are not ore than 6, here see again the area (3.). By this fact and the density of the ratios c 1 /χ h of Ml i in the interval J := [4, 8], we are able to find infinitely any pairwise non-diffeoorphic siply connected nonspin syplectic 4-anifolds Nl i such that Y < 0 and, on each of N l i, there exists no non-singular solution of the noralized Ricci flow, and oreover, Ml i and N l i are both non-spin and have the sae (χ h,c 1). Freedan s classification [11] iplies that they ust be hoeoorphic. However, each of Ml i is not diffeoorphic to any N l i because, on each of Ml i, a non-singular solution exists and, on the other hand, no non-singular solution exists on each of Nl i. Therefore, we are able to conclude that, for every natural nuber l, there exists a siply connected topological non-spin 4-anifold l satisfying the desired properties.

916 M. ISHIDA REFERENCES [1] K. Akutagawa, M. Ishida, and C. LeBrun, Perelan s invariat, Ricci flow, and the Yaabe invariants of sooth anifolds, Arch. Math., 88 (007), pp. 71 76. [] T. Aubin, Équations différentielles non linéaires et problèe de Yaabe concernant la courbure scalaire, J. Math. Pures Appl., 55:9 (1976), pp. 69 96. [3] S. Bauer and M. Furuta, Stable cohootopy refineent of Seiberg-Witten invariants: I, Invent. Math., 155 (004), pp. 1 19. [4] S. Bauer, Stable cohootopy refineent of Seiberg-Witten invariants: II, Invent. Math., 155 (004), pp. 1 40. [5] S. Bauer, Refined Seiberg-Witten invariants, Different faces of geoetry, 1 46, Int. Math. Ser. (N.Y.), Kluwer/Plenu, New York, 004. [6] V. Braungardt and D. Kotschick, Einstein etrics and the nuber of sooth structures on a four-anifold, Topology, 44 (005), pp. 641 659. [7] H.-D. Cao, Deforation of Kähler etrics to Kähler-Einstein etrics on copact Kähler anifolds, Invent. Math., 81 (1985), pp. 359 37. [8] H.-D. Cao and B. Chow, Recent developents on the Ricci flow, Bull. Aer. Math. Soc. (N.S), 36:1 (1995), pp. 59 74. [9] S. K. Donalsdon, An application of gauge theory to four-diensional topology, J. Differential Geo., 18 (1983), pp. 79 315. [10] F. Fang, Y. Zhang, and Z. Zhang, Non-singular solutions to the noralized Ricci flow equation, Math. Ann, 340 (008), pp. 647 674. [11] M. Freedan, On the topology of 4-anifolds, J. Differential Geo., 17 (198), pp. 357 454. [1] R. E. Gopf, A new construction of syplectic anifolds, Ann. of Math., 14: (1995), pp. 57 595. [13] M. Groov and H. B. Lawson, The classification of siply connected anifolds of positive scalar curvature, Ann. of Math., 111 (1980), pp. 43 434. [14] R. Hailton, Three anifolds with positive Ricci curvature, J. Differential Geo., 17 (198), pp. 55 306. [15] R. Hailton, Non-singular solutions of the Ricci flow on three-anifolds, Co. Anal. Geo., 7 (1999), pp. 695 79. [16] N. Hitchin, On copact four-diensional Einstein anifolds, J. Differential Geo., 9 (1974), pp. 435 44. [17] M. Ishida and C. LeBrun, Spin anifolds, Einstein etrics, and differential topology, Math. Res. Lett., 9 (00), pp. 9 40. [18] M. Ishida and C. LeBrun, Curvature, connected sus, and Seiberg-Witten theory, Co. Anal. Geo., 11 (003), pp. 809 836. [19] M. Ishida, The noralized Ricci flow on four-anifolds and exotic sooth structures, ariv:0807.169 [ath.dg] (008). [0] B. Kleiner and J. Lott, Notes on Perelan s papers, ariv: ath.dg/0605667 (006). [1] O. Kobayashi, Scalar curvature of a etric of unit volue, Math. Ann., 79 (1987), pp. 53 65. [] P. B. Kronheier, Minial genus in S 1 M, Invent. Math., 135 (1999), pp. 45 61. [3] C. LeBrun, Einstein etrics and Mostow rigidity, Math. Res. Lett., (1995), pp. 1 8. [4] C. LeBrun, Polarized 4-anifolds, extreal Kähler etrics, and Seiberg-Witten theory, Math. Res. Lett., (1995), pp. 653 66. [5] C. LeBrun, Four-anifolds without Einstein etrics, Math. Res. Lett., 3 (1996), pp. 133 147. [6] C. LeBrun, Kodaira diension and the Yaabe proble, Co. Anal. Geo., 7 (1999), pp. 133 156. [7] C. LeBrun, Ricci curvature, inial volues, and Seiberg-Witten theory, Invent. Math., 145 (001), pp. 79 316. [8] C. LeBrun, Einstein etrics, four-anifolds, and differential topology, in Survey in differential geoetry, Vol. VIII (Boston, MA, 00), Int. Press, Soerville. MA. 003, pp. 35 55. [9] C. LeBrun, Four-anifolds, curvature bounds, and convex geoetry, Rieannian topology and geoetric structures on anifolds, ppp. 119 15, Progr. Math., 71, Birkhauser Boston, Boston, MA, 009. [30] J. Lee and T. Parker, The Yaabe proble, Bull. A. Math. Soc., 17 (1987), pp. 37 91. [31] J. W. Morgan, T. S. Mrowka, and Z. Szabó, Product forula along T 3 for Seiberg-Witten invariants, Math. Res. Lett., 4 (1997), pp. 915 99. [3] J. W. Morgan, Z. Szabó, and C. H. Taubes, A product forula for the Seiberg-Witten invariants anf the generalized Tho conjecture, J. Differential Geo., 44 (1996), pp. 706 788.

RICCI FLOW AND SMOOTH STRUCTURES 917 [33] L. Nicolaescu, Note on Seiberg-Witten theory, Graduate Studies in Matheatics, 8 (000). [34] G. Perelan, Ricci flow with surgery on three-anifolds, ariv: ath.dg/0303109 (003). [35] M. Salvetti, On the nuber of non-equivalent differentiable structures on 4-anifolds, Manuscr. Math., 63 (1989), pp. 157 171. [36] R. Schoen and S.-T. Yau, On the structure of anifolds with positive scalar curvature, Manuscripta. Math., 8 (1979), pp. 159 183. [37] R. Schoen, Conforal deforation of a Rieannian etric to constant scalar curvature, J. Differential Geo., 0 (1984), pp. 478 495. [38] R. Schoen, Variational theory for the total scalar curvature functional for Rieannian etrics and related topics, Lec. Notes Math., 1365 (1987), pp. 10 154. [39] C. H. Taubes, The Seiberg-Witten invariants and syplectic fors, Math. Res. Lett., 1 (1994), pp. 809 8. [40] J. A. Thorpe, Soe rearks on the Gauss-Bonnet forula, J. Math. Mech., 18 (1969), pp. 779 786. [41] N. Trudinger, Rearks concerning the conforal deforation of etrics to constant scalar curvature, Ann. Scuola Nor. Sup. Pisa., (1968), pp. 65 74. [4] E. Witten, Monopoles and four-anifolds, Math. Res. Lett., 1 (1994), pp. 809 8. [43] H. Yaabe, On the deforation of Rieannian structures on copact anifolds, OsakaMath. J., 1 (1960), pp. 1 37.

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