TY - JOUR
T1 - Normalized RICCI flow, surgery, and seiberg-witten invariants
AU - Ishida, Masashi
N1 - Funding Information:
The author would like to express deep gratitude to Claude LeBrun for his constant encouragement. The author also would like to express thanks to Chanyoung Sung for the exchange of e-mails at an early stage of this work. The author is grateful to the Max-Plank-Institut für Mathematik in Bonn for its hospitality. This work is supported in part by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 20540090.
PY - 2014/2
Y1 - 2014/2
N2 - We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg-Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's λ̄ invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann. 311 (1998) 533-547] as a special case.
AB - We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg-Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's λ̄ invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann. 311 (1998) 533-547] as a special case.
KW - Ricci flow
KW - Seiberg-Witten invariants
KW - surgery
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U2 - 10.1142/S0129167X14500050
DO - 10.1142/S0129167X14500050
M3 - Article
AN - SCOPUS:84896350506
VL - 25
JO - International Journal of Mathematics
JF - International Journal of Mathematics
SN - 0129-167X
IS - 2
M1 - 1450005
ER -