  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.
