## 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, 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_{g1} (M) ≥ Vol_{g0} (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 [-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.

Original language | English |
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Pages (from-to) | 191-210 |

Number of pages | 20 |

Journal | Frontiers of Mathematics in China |

Volume | 2 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2007 |

## Keywords

- Perelman's λ-functional
- Ricci-flow
- Seiberg-Witten equations