Perelman λ泛函与Seiberg-Witten方程
Perelman's λfunctional and the Seiberg-Witten equations
摘要：
本文利用Seiberg-Witten方程研究了4维黎曼流形M上Perelman λ-泛函{λ}_M（g）的上确界.证明了：（i）若g_{1}是M上的黎曼度量，且λ_{M}（g_{1}）= λ_{M}（g_{0}），则λ_{g_{1}}（M）\geq ∈_{g_{0}}（M）.此外，等式成立当且仅当g_{1}也是具有负数量曲率的Kähler-Einstein度量。(ii)若g_{t}，t \in [-1，1]是M上的Einstein度量族，初始度量为g_{0}，则g_{t}是具有负数量曲率的Kähler-Einstein度量.
英文摘要：
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.
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