Related papers: $\cF$-functional and geodesic stability

Conformally K\"{a}hler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in K\"{a}hler geometry. We introduce uniform K-stability for toric K\"{a}hler manifolds, and show that uniform K-stability is…

This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this…

In this paper, we study the behavior of Ricci flows on compact orbifolds with finite singularities. We show that Perelman's pseudolocality theorem also holds on orbifold Ricci flow. Using this property, we obtain a weak compactness theorem…

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast…

We construct gradient K\"ahler-Ricci solitons on Ricci-flat K\"ahler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The…

In this paper, we build connections between K\"ahler-Ricci shrinkers, i.e., complete (possibly non-compact) shrinking gradient K\"ahler-Ricci solitons, and algebraic geometry. In particular, we (1). prove that a K\"ahler-Ricci shrinker is…

In this note, we show that the solution of K\"ahler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits…

We show that the existence of constant scalar curvature K\"ahler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the…

We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the K\"ahler-Ricci soliton modulo automorphisms. In our results, we do not assume that the…

We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show that geometric…

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with…

We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci…

Let $(X,\omega)$ be a compact normal K\"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical K\"ahler metrics, we introduce a geodesic…

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays,…

We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to…

In this paper, we prove that a sequence of weak almost K\"ahler-Ricci solitons under further suitable conditions converge to a K\"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As…

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such…

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively…

We prove that the existence of a Kahler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci curvature.…