Related papers: Modified extremal K\"{a}hler metrics and multiplie…
We give a complete list of non-isometric bidimensional rotation invariant K\"ahler-Einstein submanifolds of a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves in the aforementioned case a…
We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on…
The Einstein-Maxwell equations on a smooth compact 4-manifold are reformulated as a purely Riemannian variational problem analogous to Calabi's variational problem for extremal Kahler metrics. Next, Seiberg-Witten theory is used to show…
We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a…
We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal…
An almost K\"ahler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal K\"ahler metric in the sense of Calabi [8]. We observe…
Given a (possibly non-K\"ahler) Calabi--Yau threefold $(X,\Omega)$, we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of $(X,\omega,\Omega)$, where $\omega$ is a Hermitian metric on $X$. The equations defining…
Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-K\"ahler manifolds, and arise independently in mathematical physics. We reinterpret this condition…
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum…
Let (X,\Omega) be a closed polarized complex manifold, g be an extremal metric on X that represents the K\"ahler class \Omega, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative…
In this note we prove that a special family of Killing potentials on certain Hirzebruch complex surfaces, found by Futaki and Ono, gives rise to new conformally K\"ahler, Einstein-Maxwell metrics. The correspondent K\"ahler metrics are…
We introduce new probabilistic and variational constructions of (twisted) K\"ahler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional…
We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal K\"ahler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using…
We introduce a large class of canonical K\"ahler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study K\"ahler--Ricci iterations of well-behaved metrics on compact and non-compact K\"ahler…
This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we…
We obtain a necessary and sufficient condition of existence of a K{\"a}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is…
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 prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow…
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 study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature,…