Related papers: On Weighted Twisted K-Energy and Its Applications

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…

In this paper, we prove that on a smooth K\"ahler manifold, the $\mathbb{G}$-coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in…

Let (X,L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1,1)-forms in the first Chern class of L,…

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…

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 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…

We prove the lower semi-continuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact K\"ahler varieties with klt singularities. Moreover, we establish the existence of singular cscK metrics on…

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic…

There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of…

In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact K\"ahler manifold $X$: this notion include constant scalar curvature K\"ahler metrics, weighted solitons, Calabi's…

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…

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…

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is…

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature K\"{a}hler metrics…

We study the geodesic convexity of various energy and entropy functionals restricted to (non-geodesically convex) submanifolds of Wasserstein spaces with their induced geometry. We prove a variety of convexity results by means of a simple…

We study the weighted constant scalar curvature K\"ahler equations on mildly singular K\"ahler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when…

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…

The purpose of this article is to study the strict convexity of the Mabuchi functional along a $C^{1,1}$-geodesic, with the aid of the $\epsilon$-geodesics. We proved the $L^2$-convergence of the fiberwise volume element of the…

We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in the previous work of Ogusu and the author with respect to all regular…

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…