Existence Problem of Fxtremal Metran and Degeneration of Balanced Metrics

(1)针对极值度量的存在性问题，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（根据Fogarty的工作，Chow-Mumford稳定性暗示了HiItert-Mumford稳定性，而对于其逆的渐近性我们甚至什么也不知道）。(2)结合爱因斯坦环面上Kahler-Einstein度规的Monge-Ampere方程，在R^<2>的有界凸C^<2>域上，我们得到了一个具有Dirichlet条件的双曲仿射球方程，对于该方程的解，我们得到了沿边界的一个非常显式的渐近展开式。(3)已知环形或Kahler -Einstein范诺流形的反正则束允许一个Ricci-flat Kahler度规在一个合适的商上诱导出Sasaki-Einstein度规。我们研究了具有Kahler-Ricci孤子的Fano流形的反正则束的类似例子。

(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.