Masashi Ishida

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

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 Perelmans 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.

Constraints on Seiberg–Witten basic classes of anti-self-dual manifolds

Abstract We shall prove new constraints on the Seiberg–Witten basic classes of anti-self-dual manifolds. As applications of the constraints, we shall prove some new vanishing theorems for the Seiberg–Witten invariants of anti-self-dual manifolds.