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

Abstract

Abstract.We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with b + = 1. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds CP 2 # CP 2 , where 5 ≤ ≤ 8. Mathematics Subject Classification (2000). Primary 53C21; Secondary 57R55.

Exotic smooth structures on $S^2\times S^2$

“…We point out that the analogue of Corollary 4 for CP 2 #mCP 2 , when m = 5, 6, 7, 8, has been proved in [13,18]. The proof of Theorem 3 is spread out in Sections 2-5.…”

“…Hence we obtain M 1 n by performing (a ′ 1 × c ′ 1 , a ′ 1 , −n) surgery on Z. We have e(Z) = 4, σ(Z) = 0, b 1 (Z) = 1, b 2 (Z) = 4, and the intersection form of Z is isomorphic to 2H with a basis given by (13) ([…”

“…We abuse the notation slightly and let A and B also denote the Poincaré duals of [Σ 2 × {w 0 }] and [{z 0 } × Σ 3 ], respectively, in H 2 (Z; Z). If SW Z (L) = 0, then by applying the adjunction inequality to four surfaces in (13), we conclude that L = rA + sB, where r and s are even integers satisfying |r| ≤ 2 and |s| ≤ 2. Since the dimension of the Seiberg-Witten moduli space for L is nonnegative, we must have L 2 = 2rs ≥ 2e(Z) + 3σ(Z) = 8.…”

“…Part (ii) follows from [14] (see the paragraph preceding Theorem 7 in [14]). Part (iii) follows from Theorem A in [13] (by setting X = M 1 n and k = 3 in the notation of [13]).…”

On Einstein Metrics on 4-Manifolds with Finite Cyclic Fundamental Group

“…Examples of pairs of homeomorphic but not diffeomorphic simply connected manifolds such that one manifold admits an Einstein metric while the other does not were first found by Kotschick [Kot98]. Later on, other examples were constructed by LeBrun, Kotschick, the two in joint work with collaborators [IsLe02,IsLe03,BrKo05] and by others [RaSu09,IsRaSu09] in order to exhibit the existence or non existence of Einstein metrics for infinitely many homeomorphic non diffeomorphic smooth structures on the same topological space.…”