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) 针对极值度量的存在问题，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稳定性意味着HiItert-Mumford稳定性）由 Fogarty 的工作得出，而关于它的逆过程我们甚至渐进地一无所知）。(2) 与 Monge-Ampere 方程相关爱因斯坦复曲面上的卡勒-爱因斯坦度量，我们有一个双曲仿射球方程，其狄利克雷条件定义在 R^<2> 的有界凸 C^<2> 域上，对于方程的解，我们得到一个非常沿边界显式渐近展开。(3) 众所周知，环面或卡勒爱因斯坦法诺流形的反正则束承认 Ricci 平坦卡勒度量，从而导出合适商的佐佐木-爱因斯坦度量。我们研究了带有卡勒-里奇孤子的法诺流形反规范束的类似例子。

项目成果

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

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

Extremal metrics and stabilities on polarized manifolds

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

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

An affine sphere equation associated to Einstein toric surfaces

与爱因斯坦复曲面相关的仿射球方程

• 发表时间: 2008
• 期刊: Proc.4th ICCM, 2007, Hangzhou 1
• 作者: Li;L.;桐原 聡秀;T.Mabuchi
• 通讯作者: T.Mabuchi