Related papers: Weighted cscK metric (II): the continuity method

Let $X$ be a compact K\"ahler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature K\"ahler metrics on non-compact K\"ahler manifold $X-S$ with Poincar\'e--Mok--Yau asymptotic…

We establish a version, formulated in terms of non-Archimedean pluripotential theory, of the Yau-Tian-Donaldson conjecture for constant scalar curvature and, more generally, weighted extremal K\"ahler metrics with prescribed compact…

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 study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations…

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation…

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…

The notion of weighted extremal K\"ahler metrics extends the classical notion of Calabi's extremal K\"ahler metrics, but includes many well-studied objects in K\"ahler geometry such as K\"ahler-Ricci solitons and Sasaki-Einstein metrics. In…

We study the existence of weighted extremal K\"ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\"ahler manifold of constant scalar…

We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal K\"ahler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the…

We study weighted constant scalar curvature K\"ahler metrics, introduced by Lahdili as $(v,w)$-CSCK metrics, on Fano manifolds and their relationship with soliton-type metrics. In this paper, we introduce a weight function $g(v,w)$…

We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.

We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs…

Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a K\"ahler metric $\omega$ and a closed $(1, 1)$-form $\alpha$. Assuming a uniform estimate for…

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar…

The main purpose of this work is to explore the existence of constant scalar curvature Sasaki metrics in the Sasaki cone of the join of two regular Sasaki manifolds, $M_1$ and $M_2$. Furthermore, we consider some cases of continuous…

Consider a compact K\"ahler manifold X with a simple normal crossing divisor D, and define Poincar\'e type metrics on X\D as K\"ahler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature…

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…

Suppose that there exist two K\"ahler metrics $\omega$ and $\alpha$ such that the metric contraction of $\alpha$ with respect to $\omega$ is constant, i.e. $\Lambda_{\omega} \alpha = \text{const}$. We prove that for all large enough $R>0$…

We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of K\"ahler metrics with constant scalar curvature (csck) on compact K\"ahler…

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is…