Related papers: On the Critical Points of the E_k Functionals in K…

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar…

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and…

We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are…

Any $6$-dimensional strict nearly K\"ahler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we…

We use the formalism of the R{\'e}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality…

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular K\"ahler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X…

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a…

The purpose of this article is to provide a general overview of curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first prove that the space of…

A holomorphy potential is a complex valued function whose complex gradient, with respect to some K\"ahler metric, is a holomorphic vector field. Given $k$ holomorphic vector fields on a compact complex manifold, form, for a given K\"ahler…

We study rigidity on certain K\"ahler manifolds with nonnegative Ricci curvature. Among others things, we show that a complete noncompact K\"ahler surface with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature…

In this paper we consider left-invariant pseudo-K\"{a}hler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated…

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…

We investigate the asymptotic geometry of Hermitian non-K\"ahler Ricci-flat metrics with finite $\int|Rm|^2$ at infinity. Specifically, we prove: 1. Any such metric is asymptotic to an ALE, ALF-A, AF, skewed special Kasner, ALH* model at…

We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic…

We prove that a complete K\"ahler manifold with holomorphic curvature bounded between two negative constants admits a unique complete K\"ahler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly…

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce…

For elliptic principal bundles $\pi:X\ra B$ over K\"ahler manifolds it was shown by Blanchard that $X$ has a K\"ahler metric if and only both Chern classes (with real coefficients) of $\pi$ vanish. For some elliptic principal bundles, when…

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than…

We prove that on one K\"{a}hler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical K\"{a}hler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also…

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of…