Existence Problem of Fxtremal Metran and Degeneration of Balanced Metrics

极值Metran的存在问题和平衡度量的退化

• 批准号: 16204006
• 负责人: MABUCHI Toshiki
• 金额: $ 17.8万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16204006/
• 关键词: geometry; extremal metric; stability; balanced metric; degeneration; constant scalar curvature; test configuration; obstruction; Test configuration; Hitchin-Kobayashi対応; 定スカラー曲率ケーラー計量; 漸近安定性; Chow-mumford安定性; Hilbert-mumford安定性; extremal-Kah

(1) Related to the existence problem of extremal metrics, we studied various kinds of stabilities for manifolds For instance, we succeeded in showing that Chow-Mumford stability and Hilbert-Mumford stability are asymptotically equivalent (Chow-Mumford stability implies HiItert-Mumford stability by the work of Fogarty, while nothing was known even asymptotically about its converse).(2) Associated to the Monge-Ampere equation for the Kahler-Einstein metric on an Einstein toric surface, we have a hyperbolic affine sphere equation with Dirichlet condition defined on a bounded convex C^<2> domain in R^<2>, and for the solution of the equation, we obtain a very explicit asymptotic expansion along the boundary.(3) It is known that the anticanonical bundle of a toric or Kahler Einstein Fano manifold admits a Ricci-flat Kahler metric inducing a Sasaki-Einstein metric on a suitable quotient. We studied analogous examples for anticanonical bundles of Fano manifolds with Kahler-Ricci solitons.

(1)与极值度量的存在性问题相关，我们研究了流形的各种稳定性。例如，我们成功地证明了Chow-Mumford稳定性和Hilbert-Mumford稳定性是渐近等价的（Chow-Mumford稳定性通过Fogarty的工作暗示了Hilbert-Mumford稳定性，而关于它的匡威则一无所知）。(2)结合Einstein复曲面上Kahler-Einstein度规的Monge-Ampere方程，在R^中的有界凸C^域上定义了一个具有Dirichlet条件的双曲仿射球方程<2>，<2>并得到了方程解沿边界的一个非常显式的渐近展开式沿着. (3)已知复曲面或Kahler Einstein Fano流形的反正则丛允许Ricci平坦的Kahler度量在适当的商上诱导Sasaki-Einstein度量。我们研究了具有Kahler-Ricci孤子的Fano流形的反正则丛的类似例子。

项目成果

Extromal metrics and stabilities on polarized manifolds

极化流形上的极值度量和稳定性

• 发表时间: 2006
• 期刊: ICM 2006, European Mathematical Society vol.2
• 作者: 片岡瑠美子;中島昭子;他;Toshiki Mabuchi
• 通讯作者: Toshiki Mabuchi

Kahler-Ricci solitons in Sasakian geometry

萨萨基几何中的卡勒-里奇孤子

• 发表时间: 2008
• 作者: Hosokawa M;et. al.;T.Mabuchi
• 通讯作者: T.Mabuchi

Flat fronts in hyperbolic 3-space and their caustics

• DOI: 10.2969/jmsj/1180135510
• 发表时间: 2005-11
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: M. Kokubu;W. Rossman;M. Umehara;Kotaro Yamada

An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I

• DOI: 10.1007/s00222-004-0387-y
• 发表时间: 2004-10
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: T. Mabuchi

Extremal metrics and stabilities on polarized manifolds

• DOI: 10.4171/022-2/39
• 发表时间: 2006-03
• 作者: T. Mabuchi