目前Fano情形的丘-田-Donaldson猜想（Kähler-Einstein度量的存在性与流形的K稳定性等价）已被证明。剩下的一个重要研究对象是非稳定的Fano流形。本项目有三个目标：(1)对非稳定的Fano流形建立一个moment-weight型等式，其联系了丁能量的下确界和丁不变量的上确界。我们计划通过对丁泛函的发散梯度流构造渐近测地线来证明它；(2)定义与丁能量的临界点——Mabuchi度量相匹配的相对丁稳定性，以及一致的相对丁稳定性并证明其为Mabuchi度量存在的必要条件；(3)证明toric Fano流形在相对丁稳定但非一致的情形下有奇异的Mabuchi度量存在。上述目标如果达成，将会增进我们对非稳定的Fano流形的了解，以及为建立关于Mabuchi度量的丘-田-Donaldson对应打下基础。

The most important question in Kähler geometry is the existence of canonical metrics on Kähler manifolds. The central conjecture is Yau-Tian-Donaldson conjecture which predicts the equivalence of the existence of cscK metrics and the K-polystability of polarized manifolds. Since that the conjecture in the Fano case have been resolved recently, thus one of the remaining questions is how to understand the unstable Fano manifolds. The first aim of this project is establishing a moment-weight type equality for Fano manifolds which connects the infimum of Ding energy with the supremum of normalized Ding invariant. We plan to prove this equality through constructing an asymptotic geodesic ray of the divergent gradient flow of Ding functional. The second aim is to build the theory of relative Ding stability which is expected to be corresponding to Mabuchi metrics (as the critical points of Ding energy). Moreover the notion of uniformly relative Ding stable and show that it is necessary for the existence of Mabuchi metrics. The third aim is to prove the existence of some singular Mabuchi metrics on the toric Fano manifolds in the situation of relative Ding stable but not uniformly.