Perelman’s λ-functional and Seiberg-Witten equations

Abstract

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, g₀) with negative scalar curvature, (i) if g₁ is a Riemannian metric on M with λM(g1) = λM(g0), then Vol_g1(M) "e Vol_g0 (M). Moreover, the equality holds if and only if g1 is also a Kãhler-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 Kãhler-Einstein metric with negative scalar curvature.