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 Volg1 (M) ≥ Volg0 (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. © Higher Education Press 2007.