Yan Li

Let $G$ be a connected, complex reductive group. In this paper, we classify $G\times G$-equivariant normal $\mathbb{R}$-test configurations of a polarized $G$-compactification. Then for $\mathbb{Q}$-Fano $G$-compactifications, we express the H-invariant of its equivariant normal $\mathbb{R}$-test configurations in terms of the combinatory data. Based on, we compute the semistable limit of a K-unstable Fano $G$-compactification. As an application, we show that for the two K-unstable Fano $S{O}_4(\mathbb{C})$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.

In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.

In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of ${\mathrm{SO}}_4(\mathbb{C})$ and one Fano compactification of ${\mathrm{Sp}}_4(\mathbb{C})$, on which the Kähler-Ricci flow will develop singularities of type II. To the authors' knowledge, these are the first examples of Ricci flow with singularities of type II on Fano manifolds in the literature.

In this paper, we introduce a class of Sasaki manifolds with a reductive $G$-group action, called $G$-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse $G$-Sasaki Einstein metrics. A similar result is also obtained for transverse $G$-Sasaki Ricci solitons. As an application, we construct several examples of $G$-Sasaki Ricci solitons by an established openness theorem for transverse $G$-Sasaki Ricci solitons.