Perelman's λfunctional and the Seiberg-Witten equations

Abstract

In this paper we study the supremum of Perelman's λ-functional {λ}_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others 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_{g_{1}}(M)\geq Vol_{g_{0}}(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\in [-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.