Related papers: On the strict convexity of the K-energy

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a…

This article is an expository introduction to our paper Convexity of the K-energy and Uniqueness of Extremal metrics. We present the main ideas behind the proof that Mabuchi's K-energy functional is convex along weak geodesics in the space…

We show that the Mabuchi energy of any polarized manifold (X,L) is (bounded below) proper on the full space of Kahler metrics in the first Chern class of L if and only if (X,L) is asymptotically (semi)stable. In particular it now follows…

In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo…

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K \times…

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 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…

We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal K\"ahler metrics on a compact K\"ahler manifold introduced in our previous work. This extends a result by…

Donaldson conjectured \cite{Dona96} that the space of K\"ahler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and…

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{\"a}hler geometry on these varieties, with…

Let $(X,\omega)$ be a compact connected K\"ahler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of K\"ahler potentials $\mathcal H_\omega$ with respect to the $L^p$-type path length metric $d_p$. First, we…

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…

Let (X,L) be a polarized K\"ahler manifold that admits an extremal K\"ahler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is…

We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature,…

In this paper, we make a generalization of the results in \cite{Li22a} to the singular and weighted setting. In particular, we show that on a polarized projective klt variety, the $\mathbb{G}$-uniform weighted K-stability for models implies…

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study the space of generalized K\"ahler (GK) structures defined by a symplectic form $F\in \alpha$…

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified…

Based on Donaldson's method, we prove that, for an integral Kahler class, when there is a Kahler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.

In this note, we shall prove geodesic convexity of the space of K\"ahler potentials on an ALE K\"ahler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness…

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds…