Perelman’s λ-functional and Seiberg-Witten equations

doi:10.1007/s11464-007-0014-5

Front. Math. China

2007, Vol. 2

Issue (2) : 191-210 https://doi.org/10.1007/s11464-007-0014-5

Perelman’s λ-functional and Seiberg-Witten equations

FANG Fuquan, ZHANG Yuguang

Department of Mathematics, Capital Normal University, Beijing 100037, China;

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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 λM(g1) = λM(g0), then Vol_g1(M) "e Vol_g0 (M). Moreover, the equality holds if and only if g1 is also a Kãhler-Einstein metric with negative scalar curvature. (ii) If {gt}, t " [-1, 1], is a family of Einstein metrics on M with initial metric g0, then gt is a Kãhler-Einstein metric with negative scalar curvature.

Issue Date: 05 June 2007

Cite this article:

FANG Fuquan,ZHANG Yuguang. Perelman’s λ-functional and Seiberg-Witten equations[J]. Front. Math. China, 2007, 2(2): 191-210.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-007-0014-5
https://academic.hep.com.cn/fmc/EN/Y2007/V2/I2/191

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