Related papers: Extremal K\"ahler metrics

We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the…

In a previous paper, we showed that the blowup of a weighted extremal K\"ahler manifold at a relatively stable fixed point admits a weighted extremal metric. Using this result, we prove that a weighted extremal manifold is relatively…

The Wu--Yau theorem asserts that a compact K\"ahler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and…

We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains…

In the 1980s, Eugenio Calabi introduced the concept of {\it extremal K\" ahler metrics} as critical points of the $L^2$-norm functional of scalar curvature in the space of K\" ahler metrics belonging to a fixed K\"ahler class of a compact…

The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in…

We discuss the existence and non-existence of constant scalar curvature, as well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang manifold over the admissible projective bundles over local products of non-negative…

We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat…

In this new version, we correct some typos. For the readers' convenience, we also added some footnotes and more details for certain lemmas and theorems.

We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique K\"ahler-Einstein metric on such a manifold implies uniform Ding stability. The main new…

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with…

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of…

In previous work, Darvas-George-Smith obtained inequalities between the large scale asymptotic of the $J$ functional with respect to the $d_1$ metric on the space of toric K\"ahler metrics/rays. In this work we prove sharpness of these…

In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erd\H{o}s-Stone-Bollob\'as theorem, several stability…

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…

A new class of Riemannian metrics, called octonionic K\"ahler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of K\"ahler metrics on complex manifolds and of HKT-metrics of…

Some curvature properties of Kahler manifolds of indefinite metrics are studied. Analogues of a Kulkarni's theorem are proved for such manifolds.

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on K\"ahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex…

This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's…

We investigate the relationship between stability and the existence of extremal K\"ahler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a…