Andrea Loi, Fabio Zuddas
An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.
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In this paper we address two problems concerning a family of domains $M_Ω(μ) \subset \C^n$, called Cartan-Hartogs domains, endowed with a natural Kaehler metric $g(μ)$. The first one is determining when the metric $g(μ)$ is extremal (in the sense of Calabi), while the second one studies when the coefficient $a_2$ in the Engliš expansion of Rawnsley $ε$-function associated to $g(μ)$ is constant.
The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain $Ω^{B}(μ)$ endowed with the natural Kähler metric $g(μ),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(Ω^{B}(μ), g(μ))$ is constant on $Ω^{B}(μ)$ if and only if $(Ω^{B}(μ), g(μ))$ is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tenso…
An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then $(D_F, g_F)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_F$ such that $(g_F, X)$ is a Kaehler-Ricci soliton.
An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.
This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $…
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a Kähler metric $αg(μ;ν)$ $(α>0)$ on $D_{n,m}(μ)$, where $g(μ;ν)$ is the Kähler metric associated with the Kähler potential $Φ(z,w):=μν{\Vert z\Vert}^{2}-\ln(e^{-μ{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($ν>-1$) on $D_{n,m}(μ)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of th…
The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ is defined as the Hartogs type domain constructed over the product $\prod_{j=1}^kΩ_j$ of irreducible bounded symmetric domains $Ω_j$ $(1\leq j \le…
The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain $Ω^{B^{d_0}}(μ)$ endowed with the canonical metric $g(μ)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_α$ of square integrable holomorphic functions on $(Ω^{B^{d_0}}(μ), g(μ))$ with the weight $\exp\{-αφ\}$ (where $φ$ is a globally defined Kähler potential for $g(μ)$) for $α>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\vareps…
On the Bergman metric of Cartan-Hartogs domains
本文在 Cartan-Hartogs 域上引入 Bergman 对偶，证明其双全纯等价于单位球、Bergman 度量是 Kahler-Ricci 孤子及对偶有限射影诱导三者等价，并与其他典范度量比较，提出开问题。
Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains
本文证明了在覆盖有界同构域的Hartogs域上Bergman度量的刚性定理，展示了Bergman度量为Kähler-Einstein度量、同构、与单位球体双全纯等条件的等价性，回答了Yau的问题。
Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$ associated to the \K form $ω_F = -\frac{i}{2} \partial \bar{\partial} \log (F(|z_0|^2) - \|z\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that…
In this paper we study Kaehler manifolds that are strongly not relative to any projective Kaehler manifold, i.e. those Kaehler manifolds that do not share a Kaehler submanifold with any projective Kaehler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kaehler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relativ…
On weighted extremal Kähler metrics
本文探讨带权极值Kähler度量的理论发展，回顾了其与Kähler-Ricci溶菌和Sasaki-Einstein度量的关系，并总结了关于YTD猜想的最新研究成果。
Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction
本文综述了Mabuchi K-能量泛函在Kähler势空间沿弱测地线凸的证明思想，以及由此导出的常标量曲率Kähler度量和极值度量的唯一性，重点介绍概念框架与关键技术。
We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
The space of germs of extremal K\" ahler metrics in one dimension comprises three distinct ${\Bbb R}^3$ components
本文定义了局部极值Kähler度量，并证明所有一维局部极值、非Einstein Kähler度量芽的空间由三个微分同胚于${\Bbb R}^3$的分支组成。
Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface
本文在伪Hirzebruch面这类极小直纹面上证明每个凯勒类都存在高阶极值凯勒度量，并利用顶端Bando-Futaki不变量证明任何凯勒类中都不存在高阶常数量曲率凯勒度量。
From Calabi's extremal metrics to scalar-flat Kähler cones
本文证明了在特定条件下，复锥体 admits 标量平坦的Kähler锥度量，回答了关于极值Sasaki流形与Sasaki球面的标量曲率问题。
We generalize the bifurcation technique of Bando-Mabuchi in the context of extremal metrics.
We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.
We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.
In joint work with Chen and Weber, the author has elsewhere shown that CP2#2(-CP2) admits an Einstein metric. The present paper gives a new and rather different proof of this fact. Our results include new existence theorems for extremal Kahler metrics, and these allow one to prove the above existence statement by deforming the Kahler-Einstein metric on CP2#3(-CP2) until bubbling-off occurs.
In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.
This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.
The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.
The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces
Griffiths extremality, interpolation of norms, and Kähler quantization
本文引入了 Griffiths 极值 Finsler 度量，证明其由复 Monge--Ampère 型方程刻画并求解了 Dirichlet 问题，同时展示了其在 Kähler 几何 PDE 解量子化中的应用。
Weighted extremal metrics on blowups
本文证明紧致凯勒流形若容许环面作用的加权极值度量，则其在相对稳定点处的爆破也容许加权极值度量，从而将极值度量结果推广到极值 Sasaki 度量、凯勒–里奇孤立子形变及 μ-cscK 度量等典范度量。
We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper.
Modified extremal Kähler metrics and multiplier Hermitian-Einstein metrics
本文受Mabuchi的$σ$型乘子厄米-爱因斯坦度量启发，提出$σ$-极值凯勒度量概念，推广Calabi极值凯勒度量，并证明在Fano流形上乘子厄米-爱因斯坦度量的存在性蕴含$σ$-极值凯勒度量的存在性。
Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) Kähler metric, possibly of Poincaré type, on every component of D. We also show that when the divisor is smooth, the constant scalar curvature/extremal metric on X\D is asymptotically a product near the divisor.
We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kähler metrics on a compact Kähler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy. This implies a suitable notion of weighted K-semistability of a Kähler manifold admitting a weighted extremal Kähler metric.
In this short note we show that the existence of bilaterally symmetric extremal Kähler metrics on $\mathbb{CP}^2\sharp 2\bar{\mathbb{CP}^2}$.
In this paper, we study rotationally symmetric extremal Kähler metrics on $\mathbb C^n$ ($n\geq 2$) and $\mathbb C^2 \backslash\{0\}$. We present a classification of such metrics based on the zeros of the polynomial appearing in Calabi's Extremal Equation. As applications, we prove that there are no $U(n)$ invariant complete extremal Kähler metrics on $\mathbb C^n$ with positive bisectional curvature, and we give a smooth extension lemma for $U(n)$ invariant extremal Kähler metrics on $\mathbb{C}^n\backslash\{0\}$. We retrieve known examples of smooth or singular extremal Kähler metrics on Hir…
New examples of extremal Kähler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.
In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic automorphisms of $X$. Then the polarization class $c_1(L)$ will be shown to admit an extremal Kaehler metric if $(X,L)$ is strongly K-stable relative to $T$.
We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.
In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.
Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.
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 of the Calabi energy and the extremal polynomials. In particular, we give an example where the modified K energy is bounded from below but not proper.
Balanced metrics for extremal Kähler metrics and Fano manifolds
本文先综述作者在代数几何平衡度量与稳定性概念方面的工作，再仅用Hopf--Rinow定理证明完备黎曼流形上测地凸函数临界点存在性的结论并推广至局部紧完备长度度量空间。
We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics minimizing it. More generally, a relative version of the K-energy is convex, and bounded below if there exists a Sasaki-extremal metric, providing an important necessary condition for Sasaki-extremal metrics. Another application is a proof of the uniqueness of Sasaki-extremal metr…
We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.
The authors found extremals of arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank two. They use for this the Pontryagin Maximum Principle for the corresponding time-optimal problem in coordinates of the first kind. The obtained results are applied to the case of left-invariant sub-Riemannian metric with the same distribution.
This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.
Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally maximizing. Second, we obtain a parametrization of timelike, spacelike, lightlike normal extremal trajectories by Jacobi functions. Third, a discrete symmetry group and its fixed points which are Maxwell points of of timelike and spacelike normal extremals, are described. An estimate for the cut time (the time of loss of optimality) on extremal trajectories is deri…
On the existence of coupled extremal Kähler metrics on ruled surfaces
本文研究亏格2黎曼曲面上极小直纹曲面的耦合极值Kähler度量存在性，通过Calabi ansatz将方程化为常微分方程，给出参数区域刻画，并证明其包含经典极值度量情形。
Extremal metrics involving scalar curvature
研究标量曲率刚性定理成立的极值度量，给出度量非刚性的充分条件并构造黎曼流形实例。
We define K-stability of a polarized Sasakian manifold relative to a maximal torus of automorphisms. The existence of a Sasaki-extremal metric in the polarization is shown to imply that the polarization is K-semistable. Computing this invariant for the deformation to the normal cone gives an extention of the Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an obstruction of Sasaki-extremal metrics. We use this to give a list of examples of Sasakian manifolds whose Sasaki cone contains no extremal representatives. These give the first examples of Sasaki cones of dimensi…
We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.
We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $φ_0$ close to $ω$, the Calabi flow starting at $φ_0$ exists for all time and the modified Calabi flow starting at $φ_0$ will always be close to $ω$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $ω$ exponentially fast.
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 of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki--Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kahler metric, illuminating and uniting several explicit constructions in Kahler and Sasaki geometry. It also…
In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle.
On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, by moving frames, we show that any non-CSC HCMU metric can not be isometrically minimal immersed into 3-dimensional real space forms even locally. In general, any non-CSC HCMU metric can not be isometrically immersed into 3-dimensional real space forms with constant mean curvature (CMC).
We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal Kähler 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 some weighted balanced metrics.
We construct extremal metrics on the total space of certain destabilising test configurations for strictly semistable Kähler manifolds. This produces infinitely many new examples of manifolds admitting extremal Kähler metrics. It also shows for such metrics a new phenomenon of jumping of the complex structure along fibres.
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.
In this paper we introduce higher extremal Kahler metrics. We provide an example of the same on a minimal ruled surface. We also prove a perturbation result that implies that there are non-trivial examples of higher constant scalar curvature metrics, which are basically metrics where the top Chern form is harmonic. We also give a relatively short proof of a formula due to Liu for the Bando-Futaki invariants (which are obstructions for the existence of harmonic Chern forms) of hypersurfaces in projective space.
In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.
Riemannian $λ_1$-extremal Metrics on Generalized Flag Manifolds
本文利用李理论数据给出的充要条件，研究广义旗流形上 Kähler-Einstein 度量作为 Laplace 算子第一正特征值泛函临界点的 $λ_1$-极值性，并得到偏旗流形的新分类结果。
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