On normalized Ricci flow and smooth structures on four-manifolds with b + =1

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1 Arch. ath. 92 (2009, c 2009 Birkhäuser Verlag Basel/Switzerland 0003/889X/ , published online DOI 0.007/s Archiv der atheatik On noralized Ricci flow and sooth structures on four-anifolds with b + = asashi Ishida, Rareş Răsdeaconu, and Ioana Şuvaina Abstract. We find an obstruction to the existence of non-singular solutions to the noralized Ricci flow on four-anifolds with b + =. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the noralized Ricci flow and exotic sooth structures on the topological 4-anifolds CP 2 #lcp 2, where 5 l 8. atheatics Subject Classification (2000. Priary 53C2; Secondary 57R55. Keywords. Noralized Ricci flow, exotic sooth structures, Seiberg-Witten invariant.. Introduction. Let X be a closed oriented Rieannian anifold of diension n 3. The noralized Ricci flow on X is the following evolution equation: t g = 2Ric g + 2 ( X s gdµ g n X dµ g, g where Ric g,s g are the Ricci curvature and the scalar curvature of the evolving Rieannian etric g and dµ g is the volue easure with respect to g. Recall that a one-paraeter faily of etrics {g(t}, where t [0,T for soe 0 <T,is called a solution to the noralized Ricci flow if it satisfies the above equation at all x X and t [0,T. A solution {g(t} on a tie interval [0,T is said to be axial if it cannot be extended past tie T. In this paper we are interested in solutions which are particularly nice. The following definition was first introduced and studied by Hailton [, 6]: Definition. A axial solution {g(t}, t [0,T, to the noralized Ricci flow on X is called non-singular if T = and the Rieannian curvature tensor R g(t of g(t satisfies sup X [0, R g(t <.

2 356. Ishida, R. Răsdeaconu, and I. Şuvaina Arch. ath. Fang and his collaborators [9] pointed out that for a 4-anifold with negative Perelan invariant [23, 4], which is equivalent to the Yaabe invariant in this situation [2], the existence of non-singular solutions forces a topological constraint on the 4-anifold. On the other hand, the first author proved [2] that, in diension four, the existence of non-singular solutions is fundaentally related to the sooth structure considered. An iportant ingredient in his theores was the non-triviality of the Seiberg-Witten invariant. In the case when the underlying anifold has b + 2, this is a diffeoorphis invariant. However, when b + = the invariant depends on the choice of an orientation on H 2 (X, Z and H (X, R. The obstructions in [2, 3] are for anifolds with b + 2. We extend these results to the case b + = as follow: Theore A. Let X be a closed oriented sooth 4-anifold with b + (X =and 2χ(X +3τ(X > 0. Assue that X has a non-trivial Seiberg-Witten invariant. Then, there do not exist non-singular solutions to the noralized Ricci flow on X#kCP 2 if k> 3 (2χ(X+3τ(X. By using Theore A, we study anifolds with sall topology and ephasize how the change of the sooth structure reflects on the existence or non-existence of solutions of the noralized Ricci flow: Theore B. For 5 l 8, the topological 4-anifold := CP 2 #lcp 2 satisfies the following properties:. adits a sooth structure of positive Yaabe invariant on which there exists a non-singular solution to the noralized Ricci flow. 2. adits a sooth structure of negative Yaabe invariant on which there exist non-singular solutions to the noralized Ricci flow. 3. adits infinitely any distinct sooth structures all of which have negative Yaabe invariant and on which there are no non-singular solutions to the noralized Ricci flow for any initial etric. 2. Polarized 4-anifolds and Seiberg-Witten invariants. Let X be a closed oriented sooth 4-anifold. Any Rieannian etric g on X gives rise to a decoposition H 2 (X, R =H g + Hg, where H g +, Hg consists of cohoology classes for which the haronic representative is g-self-dual or g-anti-self-dual, respectively. Notice that b + (X :=dih g + is a non-negative integer which is independent of the etric g. In this article, we always assue that b + (X and ainly consider the case when b + (X =. For a fixed b + (X-diensional subspace H H 2 (,R on which the intersection for is positively defined, we consider the set of all Rieannian etrics g for which H g + = H is satisfied. The Rieannian etric g satisfying this property is called a H-adapted etric. Under the assuption that there is at least one H-adapted etric, H is called a polarization of X and we call (X, H a polarized 4-anifold following [9]. For any given eleent α H 2 (X, R and a

3 Vol. 92 (2009 Noralized Ricci flow, sooth structures 357 polarization H of X, we use α + to denote the orthogonal projection of α, with respect to the intersection for of X, on the polarization H. For any polarized 4-anifold (X, H, we can define a differential topological invariant [32, 9] of (X,H, by using Seiberg-Witten onopole equations [32]. We briefly recall the definition, referring to [32, 9] for ore details. Let s be a spin c structure of the polarized 4-anifold (X, H. Let c (L s H 2 (X, R be the first Chern class of the coplex line bundle L s associated to s. Suppose that d s := (c 2 (L s 2χ(X 3τ(X/4 = 0, which forces the virtual diension of the Seiberg-Witten oduli space to be zero. Let g be a H-adapted etric and assue that c + (L s 0 with respect to H = H g + is satisfied. Then [32, 9], the Seiberg-Witten invariant SW X (s,h is defined to be the nuber of solutions of a generic perturbation of the Seiberg-Witten onopole equations, odulo gauge transforation and counted with orientations. We can still define the Seiberg-Witten invariant of (X, H for any spin c structure s for which c + (L s 0 and d s is even and positive. In this case, SW X (s,h is defined as the pairing <η ds 2, [ s ] >. Here η is the first Chern class of the based oduli space as a S -bundle over the Seiberg-Witten oduli space s and [ s ] is the fundaental hoology class of s. Hence, the Seiberg- Witten invariant SW X (s,h of a polarized 4-anifold (X, H is well-defined for any spin c structure s with c + (L s 0. oreover, it is known [2] that SW X (s,h is independent of the choice of the polarization H if b + (X 2, or b + (X =and 2χ(X+3τ(X > 0. One of the crucial properties of the Seiberg-Witten invariants above is that the non-triviality of the value SW X (s,h for a spin c structure s with c + (L s 0 forces the existence of a non-trivial solution of the Seiberg-Witten onopole equations for any H-adapted etric. Using this, LeBrun [7, 20, 9] proved Theore ([7, 20]. Let (X, H be a polarized sooth copact oriented 4-anifold and let s be a spin c structure of X and let c + 0be the orthogonal projection to H with respect to the intersection for of X. Assue that SW X (s,h 0. Then, every H-adapted etric g satisfies the following bounds: s 2 gdµ g 32π 2 (c + 2, 4π 2 X X ( 2 W g s2 g dµ g (c+ 2. s g is the scalar curvature of g and W + g is the self-dual Weyl curvature of g. On the other hand, let N be a closed oriented sooth 4-anifold with b + (N = 0 and k = b 2 (N. By the celebrated result of Donaldson [8], there are classes e, e 2,, e k H 2 (N,Z descending to a basis of H 2 (N,Z/torsion with respect to which the intersection for is diagonal and e 2 i = for all i. An eleent β H 2 (N,Z is called characteristic if the intersection nuber β x x x (od 2.

4 358. Ishida, R. Răsdeaconu, and I. Şuvaina Arch. ath. If β is characteristic, then β w 2 (N (od 2 and oreover there is a spin c structure t on N such that c (L t =β. Then odulo torsion, β can be written as k i= a ie i, where a i are integers. Let x := k i= x ie i, where x i are integers. Then we have β x = k i= a ix i and x x = k i= x2 i. This tells us that β is characteristic if and only if the a i are odd integers, where i =,,k.for exaple, we can obtain characteristic eleents by taking a i = ±. The following result includes Lea of [9] as a special case. Proposition 2. Let X be a closed oriented sooth 4-anifold with b + (X and 2χ(X +3τ(X > 0. oreover, suppose that the Seiberg-Witten invariant of X is non-trivial. Let N be a closed oriented sooth 4-anifold with b (N =b + (N =0. Let H be any polarization of a connected su := X#N. Then there is a spin c structure s on such that SW X (s,h 0and the self-dual part c + of the first Chern class of the coplex line bundle associated with s satisfies ( (c + 2 2χ(X+3τ(X. Proof. Notice that the Seiberg-Witten invariant of X is well-defined and independent of the choice of the polarization under the assuption that b + (X 2 or b + (X = and 2χ(X +3τ(X > 0. Suppose that c is the spin c structure on X with non-trivial Seiberg-Witten invariant. Let α := c (L c H 2 (X, Z be the first Chern class of the coplex line bundle L c associated to c. Then, the non-triviality of Seiberg-Witten invariant forces the diension d c of the Seiberg- Witten oduli space to be non-negative and we therefore have α 2 2χ(X + 3τ(X > 0. oreover, since for any given polarization of X, we have (α + 2 α 2, we obtain (2 (α + 2 2χ(X+3τ(X. Let e, e 2,, e n H 2 (N,Z be cohoology classes descending to a basis of H 2 (N,Z/torsion with respect to which the intersection for is diagonal, where n = b 2 (N. Let H be a polarization of the connected su := X#N. Choose new generators ê i = ±e i for H 2 (N,Z such that (3 α + (ê i + 0 with respect to the polarization H. Then n i= êi is a characteristic class and there is a spin c structure t on N such that c (L t = n i= êi. Notice also that we have c 2 (L t =( n i= êi 2 = n = b 2 (N. Consider the spin c structure s := c#t on the connected su := X#N. The first Chern class of the coplex line bundle associated with s satisfies c (L s = α + c (L t =α + n i= êi, where we are using the sae notations, α, c (L t, ê i, to denote the induced cohoology classes in H 2 (,Z. The gluing construction for the solutions of the Seiberg-Witten onopole equations on := X#N, as in the proof of Theore 3. in [28] (see also the proof of Proposition 2 in [6] for b + 2,

5 Vol. 92 (2009 Noralized Ricci flow, sooth structures 359 tells us that SW (s,h 0. On the other hand, we obtain the following bound on the square (c + 2 of the orthogonal projection c + of c (L s into the polarization H: ( 2 n n n (c + 2 = α + + (ê i + =(α (α + (ê i + + ((ê i + 2 i= (α + 2, where we used (3. This bound and (2 iplies the desired bound (. 3. Asyptotic behavior of the Ricci curvature of non-singular solutions. Let us recall the following result on the trace free part of the Ricci curvature of a long tie solution of the noralized Ricci flow. Lea 3 ([9, 2]. Let X be a closed oriented Rieannian n-anifold and assue that there is a long tie solution {g(t}, t [0,, to the noralized Ricci flow. Assue oreover that the solution satisfies s g(t C and (4 ŝ g(t := in s g(t(x c<0, x X where the constants C and c are independent of both x X and tie t [0,. Then, the trace-free part r g(t of the Ricci curvature satisfies (5 when. + X i= r g(t 2 dµ g(t dt 0 On the other hand, there is a natural diffeoorphis invariant arising fro a variational proble of the total scalar curvature of Rieannian etrics on any given closed oriented Rieannian anifold X of diension n 3. As was conjectured by Yaabe [33], and later proved by Trudinger, Aubin, and Schoen [4, 26, 3], every conforal class on any sooth copact anifold contains a Rieannian etric of constant scalar curvature. To be ore precise, for any conforal class [g] ={vg v : X R + }, we can consider an associated nuber Y [g], which is called the Yaabe constant of the conforal class [g], and is defined by Y [g] = inf h [g] X s h dµ h ( X dµ h n 2 n where dµ h is the volue for with respect to the etric h. It is known [4, 26, 3] that this nuber is realized as the constant scalar curvature of soe etric in the conforal class [g]. Then, Kobayashi [5] and Schoen [27] independently introduced an invariant of X by considering Y(X :=sup [g] C Y [g], where C is the set of all conforal classes on X. This is now known as the Yaabe invariant of X. We have the following bound:, i=

6 360. Ishida, R. Răsdeaconu, and I. Şuvaina Arch. ath. Lea 4 ([2]. Let X be a closed oriented Rieannian anifold of diension n 3 and assue that the Yaabe invariant of X is negative, i.e., Y(X < 0. If there is a solution {g(t}, t [0,T, to the noralized Ricci flow, then the solution satisfies the bound (4. ore precisely, the following is satisfied: ŝ g(t := in s Y(X g(t(x < 0. x X (vol g(0 2/n We recall now the following definition: Definition 2 ([2]. A axial solution {g(t}, t [0,T, to the noralized Ricci flow on X is called quasi-non-singular if T = and the scalar curvature s g(t of g(t satisfies sup X [0, s g(t <. Notice that any non-singular solution is quasi-non-singular. Proposition 5. Let X be a closed oriented sooth 4-anifold with b + (X and 2χ(X +3τ(X > 0. Assue that X has a non-trivial Seiberg-Witten invariant. Let N be a closed oriented sooth 4-anifold with b (N =b + (N =0. If there is a quasi-non-singular solution to the noralized Ricci flow on the connected su := X#N, then the trace-free part r g(t of the Ricci curvature satisfies (5 when +. Proof. First of all, notice that the connected su has non-trivial Seiberg-Witten invariant with respect to any polarization by Proposition 2. By Witten s vanishing theore [32], this iplies that cannot adit any etric of positive scalar curvature. On the other hand, it is known [8] that the Yaabe invariant of any closed n-anifold Z which cannot adit etrics of positive scalar curvature is given by 2/n (6 Y(Z = inf s g n/2 dµ g, g R Z Z where R Z is the set of all Rieannian etrics on Z. Cobining the first inequality in Theore with the inequality in Proposition 2 we get the following bound: (7 s 2 gdµ g 32π 2 (2χ(X+3τ(X. Note that this bound holds for any H-adapted etric on, where H is any polarization of. In particular, it holds for any etric g. Therefore, (6 and (7 iply Y( 4π 2(2χ(X+3τ(X < 0. This bound and Lea 4 tell us that any solution to the noralized Ricci flow on satisfies ŝ g(t 4π 2(2χ(X+3τ(X vol g(0 < 0.

7 Vol. 92 (2009 Noralized Ricci flow, sooth structures 36 This inequality cobined with Lea 3 shows that any quasi-non-singular solution to the noralized Ricci flow on ust satisfy (5 when Proof of Theore A. We are now in the position to prove Theore A, which is a special case of the following result: Theore 6. Let X be a closed oriented sooth 4-anifold with b + (X and 2χ(X+3τ(X > 0. Assue that X has a non-trivial Seiberg-Witten invariant. Let N be a closed oriented sooth 4-anifold with b (N =b + (N =0. Then, there do not exist quasi-non-singular solutions to the noralized Ricci flow on X#N if 3b 2 (N > 2χ(X+3τ(X holds. In particular, there is no non-singular solution to the noralized Ricci flow. Proof. Suppose that there would be a quasi-non-singular solution {g(t} to the noralized Ricci flow on the connected su := X#N. Then the second inequality in Theore tells us that, for any tie t, g(t ust satisfy 4π 2 X ( 2 W + g(t 2 + s2 g(t 24 dµ g(t 2 3 (c+ 2. for any spin c structure s with SW (s,h 0, where H := H + g(t. However, Proposition 2 now asserts that the connected su := X#N has a spin c structure with (c + 2 2χ(X+3τ(X. We therefore conclude that (8 4π 2 ( 2 W + g(t 2 + s2 g(t 24 dµ g(t 2 3 (2χ(X+3τ(X. On the other hand, we have the following Gauss-Bonnet like forula: 2χ(+3τ( = ( 4π 2 2 W + g(t 2 + s2 g(t 24 r g(t 2 dµ g(t. 2 In particular, we obtain 2χ(+3τ( = = + 4π 2 (2χ(+3τ( dt + ( 2 W + g(t 2 + s2 g(t 24 r g(t 2 dµ g(t dt. 2 Since Proposition 5 tells us that a quasi-non-singular solution {g(t} ust satisfy (5, by taking in the above inequality, we obtain + ( 2χ(+3τ( = li 4π 2 2 W + g(t 2 + s2 g(t (9 dµ g(t dt. 24

8 362. Ishida, R. Răsdeaconu, and I. Şuvaina Arch. ath. oreover, by the inequality (8, we get + ( 4π 2 2 W + g(t 2 + s2 g(t dµ g(t dt (2χ(X+3τ(X dt = 2 3 (2χ(X+3τ(X. This bound and (9 tell us that the following holds: 3(2χ(+3τ( 2(2χ(X+3τ(X. Since we have 2χ(+3τ( =2χ(X+3τ(X b 2 (N, we get 3(2χ(X+3τ(X b 2 (N 2(2χ(X +3τ(X, which is equivalent to 3b 2 (N 2χ(X +3τ(X. By contraposition, we obtain the desired result. 5. Proof of Theore B. Part of the proof of Theore B is based on the existence results for siply connected anifolds with b + = and aple canonical line bundle. Such exaples have only been recently found, and we state the results we need for convenience: Theore 7 ([7, 25, 22]. There exist siply connected coplex surfaces of general type, with b + =, c 2 =, 2, 3 or 4 and aple canonical bundle. Proof of Theore B. If we want to construct a sooth structure on the anifold X which has positive Yaabe invariant and adits non-singular solutions for the noralized Ricci flow, then we can just consider the canonical sooth and coplex structures of the coplex projective plane blown-up at l points, where 3 l 8. The existence of an Einstein etric is given by a faous result of Tian [30]. Hence, on these Del Pezzo surfaces, there are non-singular solutions (fixed points of the noralized Ricci flow by taking the Kähler-Einstein etrics with positive scalar curvature as initial etrics. Since the scalar curvature of these etrics is positive, Lea.5 in [5] tells us that the Yaabe invariants of these anifolds ust be also positive. In the second case of the theore, we are going to consider the sooth structures associated to the coplex structures of general type found in Theore 7. On these anifolds, a nice result of Cao [5, 6] tells us that non-singular solutions to the noralized Ricci flow exist if we start with a Kähler etric whose Kähler for is in the cohoology class of the canonical line bundle. oreover, for surfaces of general type the Yaabe invariant is strictly negative [8]. For the proof of the third part of the theore, we use the exotic structures on anifolds with sall topology constructed by Akhedov and Park []. In Section 6 of [], they exhibit infinitely any sooth structures, X i, on the topological space CP 2 #2CP 2. The sooth structures are distinguished by their Seiberg-Witten invariants. On := CP 2 #lcp 2, 5 l 8, we consider the sooth structures i := X i #(l 2CP 2,i N. Each i is hoeoorphic to CP 2 #lcp 2 as each

9 Vol. 92 (2009 Noralized Ricci flow, sooth structures 363 X i is hoeoorphic to CP 2 #2CP 2. oreover, we know that the Seiberg-Witten invariants of the anifolds X i are non-trivial and distinct. We can use the gluing forula for the connected su with copies of CP 2 to conclude that infinitely any Seiberg-Witten invariants of i reain distinct. Hence on, we have constructed infinitely any distinct sooth structures. As c 2 ( i =9 l>0, the Yaabe invariant of each sooth structure is negative [8]. The anifolds X i have c 2 (X i = 7, non-trivial Seiberg-Witten invariant by construction and of course 3(l 2 > c 2 (X i =7, as 5 l 8. Hence, Theore A tells us that there are no non-singular solutions to the noralized Ricci flow on any i for any initial etric. Acknowledgeents. We would like to express our deep gratitude to Claude LeBrun for his war encourageent. The first author is partially supported by the Grantin-Aid for Scientific Research (C, Japan Society for the Prootion of Science, No The second author would like to thank CNRS and IRA Strasbourg for support and the excellent conditions provided while this work was copleted. References [] A. Akhedov and B. Park, Exotic sooth structures on sall 4-anifolds withodd signatures, arxiv:ath/ [2] K. Akutagawa,. Ishida, and C. LeBrun, Perelan s invariant, Ricci flow, and the Yaabe invariants of sooth anifolds, Arch. ath. 88, 7 76 (2007. [3] T. Aubin, Equations du type onge-apère sur les variétés Kählériennes copactes, C. R. Acad. Sci. Paris. 283, 9 2 (976. [4] T. Aubin, Équations différentielles non linéaires et problèe de Yaabe concernant la courbure scalaire, J. ath. Pures Appl. (9 55, (976. [5] H.-D. Cao, Deforation of Kähler etrics to Kähler-Einstein etrics on copact Kähler anifolds, Invent. ath. 8, (985. [6] H.-D. Cao and B. Chow, Recent developents on the Ricci flow, Bull. Aer. ath. Soc. (N.S ( 36, (995. [7] F. Catanese and C. LeBrun, On the scalar curvature of Einstein anifolds, ath. Res. Lett. 4, (997. [8] S. K. Donalsdon, The orientation of Yang-ills oduli spaces and 4-anifold topology, J. Differential. Geo. 26, (987. [9] F. Fang, Y. Zhang, and Z. Zhang, Non-singular solutions to the noralized Ricci flow equation, ath. Ann. 340, (2008. [0]. Freedan, On the topology of 4-anifolds, J. Diff. Geo. 7, (982. [] R. Hailton, Non-singular solutions of the Ricci flow on three-anifolds, Co. Anal. Geo. 7, (999. [2]. Ishida, The noralized Ricci flow on four-anifolds and exotic sooth structures, arxiv: [ath.dg] (2008.

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11 Vol. 92 (2009 Noralized Ricci flow, sooth structures 365 [3] N. Trudinger, Rearks concerning the conforal deforation of etrics to constant scalar curvature, Ann. Scuola Nor. Sup. Pisa, 22, (968. [32] E. Witten, onopoles and four-anifolds. ath. Res. Lett., (994. [33] H. Yaabe, On the deforation of Rieannian structures on copact anifolds, Osaka ath. J. 2, 2 37 (960. [34] S.-T. Yau, Calabi s conjecture and soe new results in algebraic geoetry, Proc. Nat. Acad. USA. 74, (977. asashi Ishida, Departent of atheatics, Sophia University, 7- Kioi-Cho, Chiyoda-Ku, Tokyo , Japan e-ail: ishida@.sophia.ac.jp Rareş Răsdeaconu, Einstein Institute of atheatics, Hebrew University of Jerusale, Givat Ra, Jerusale, 9904, Israel e-ail: rares@ath.huji.ac.il Ioana Şuvaina, Courant Institute of atheatical Sciences, 25 ercer St. New York 002, USA e-ail: ioana@cis.nyu.edu Received: 3 Noveber 2008