Perelman's \lambda functional and the Seiberg-Witten equations

Abstract

In this paper we study the supremum of Perelman's \lambda-functional {\lambda }_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others that, for a compact K\"{a}hler-Einstein complex surface (M, J, g_{0}) with negative scalar curvature, (i) If g_{1} is a Riemannian metric on M with \lambda_{M}(g_{1})= \lambda_{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 K\"{a}hler-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 K\"{a}hler-Einstein metric with negative scalar curvature.