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 λ_M(g₁) = λ_M(g₀), then $Vol_{g_1 } $$ (M) ⩾ $Vol_{g_0 } $$ (M). Moreover, the equality holds if and only if g₁ 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 g₀, then g_t is a Kähler-Einstein metric with negative scalar curvature.

Keywords

Perelman’s λ-functional / Ricci-flow / Seiberg-Witten equations

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