Related papers: Geodesic rays and stability in the cscK problem

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,…

In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (cscK) metric…

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 give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted K\"ahler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability…

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity…

For the Newtonian N-body problem, we study the Jacobi-Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More…

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced)…

For the Newtonian \(N\)-body problem at nonnegative energy, we study solution sets selected by the Jacobi--Maupertuis variational principle and by the associated stationary Hamilton--Jacobi equation. We prove a compactness/stability theorem…

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,\alpha}, 0<\alpha<1$. An essential step is to establish that the rays can be extended as solutions of a…

We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T…

We prove continuity results for new stability thresholds related to uniform K-stability and deduce that uniform K-stability is an open condition in the K\"ahler cone of any compact K\"ahler manifold, thus establishing an algebro-geometric…

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…

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that…

We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space $\mathcal{E}^{1,T}(X,\omega)$, covering the case of divisors with mixed cusp and conic singularities. We then prove…

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when…

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity…

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and…

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given K\"ahler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic…

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be…

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic…