Perelman's λ-functional and Seiberg-Witten equations

In this paper, we estimate the supremum of Perelman's λ-functional λ _M (g) on Riemannian 4-manifold (M, g) by using the Seiberg-Witten equations. Among other things, we prove that, for a compact Kähler-Einstein complex surface (M, J, g ₀) with negative scalar curvature, (i) if g ₁ is a Riemannian metric on M with λ _M (g ₁) = λ _M (g ₀), then Vol_g1 (M) ≥ Vol_g0 (M). Moreover, the equality holds if and only if g ₁ is also a Kähler-Einstein metric with negative scalar curvature. (ii) If {g _t}, t [-1, 1], is a family of Einstein metrics on M with initial metric g ₀, then g _t is a Kähler-Einstein metric with negative scalar curvature.