Related papers: Remarks on Singular K\"ahler-Einstein Metrics

This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD)…

In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as…

We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight)…

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…

Solutions to vacuum Einstein field equations with cosmological constant, such as the de Sitter space and the anti-de Sitter space, are basic in different cosmological and theoretical developments. It is also well known that complex…

We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of…

We analyze the existence of K\"ahler-Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity $(X,p)$. This boils down to solve a Dirichlet problem for certain complex Monge-Amp\`ere equations. We…

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and T{\o}nnesen-Friedman), arising from a base with a local K\"ahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein…

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…

Special Ricci-Hessian equations on K\"ahler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367--380] involve functions $\tau$ on $M$ and state that, for some function $\alpha$ of the real variable $\tau$, the…

Generalizing Fujita-Odaka invariant, we define a function $\tilde{\delta}$ on a set of generalized $b$-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform $K$-stability. A key role is played by a…

We propose new types of canonical metrics on K\"ahler manifolds, called coupled K\"ahler-Einstein metrics, generalizing K\"ahler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and…

We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient K\"ahler-Ricci soliton $(M,\,g,\,X)$ with soliton metric $g$ with bounded scalar curvature $\operatorname{R}_{g}$ whose…

We focus on the classical open problem of the classification of K\"ahler-Einstein manifolds that can be K\"ahler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in…

We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and $n$-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as…

The main purpose of the present paper is to investigate the symmetry properties of a K\"ahler manifold involving the Ricci tensor. In this context, the most symmetric manifolds are K\"ahler-Einstein spaces, and their natural generalizations…

We study metric aspects of the universal moduli space of solutions to Hitchin's equations as the complex structure $J$ varies over the Teichm\"uller space $\mathcal{T}$ of a closed surface $\Sigma$. Our approach is gauge theoretical and…

Starting with a model conical K\"ahler metric, we prove a uniform scalar curvature bound for solutions to the conical K\"ahler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also…

In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain…

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the K\"ahler-Einstein…