Perelman’s λ-functional and Seiberg-Witten equations

In this paper, we estimate the supremum of Perelman’s λ-functional λM(g) on Riemannian 4-manifold (M, g) by using the Seiberg-Witten equations. Among other things, we prove that, for a compact Kahler-Einstein complex surface (M, J, g0) with negative scalar curvature, (i) if g1 is a Riemannian metric on M with λM(g1) = λM(g0), then \(Vol_{g_1 } \) (M) ⩾ \(Vol_{g_0 } \) (M). Moreover, the equality holds if and only if g1 is also a Kahler-Einstein metric with negative scalar curvature. (ii) If {gt}, t ∈ [−1, 1], is a family of Einstein metrics on M with initial metric g0, then gt is a Kahler-Einstein metric with negative scalar curvature.