Gianni Manno, Filippo Salis
We give a complete list of non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds of a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves in the aforementioned case a classical and long-staying problem addressed among others in [5] and [24].
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The object of the present paper is to study some properties of para-K$ä$hler manifold whose metric is conformal Einstein soliton. We have studied some certain curvature properties of para-K$ä$hler manifold admitting conformal Einstein soliton.
K{ä}hler Einstein manifolds and the Calabi curvature operator
本研究针对卡拉比流形上的卡拉比曲率算子，证明其满足不同正性条件时对应标量曲率、里奇曲率等的正性结论，还得出满足特定条件的紧卡拉比爱因斯坦流形必具有非负常全纯截面曲率的贡献。
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact type.
The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galicki and Piccinni in \emph{$3-$Sasakian geometry, nilpotents orbits, and exeptional quotients}. We construct explicitly our new examples as quaternion-K$\ddot{\mathrm{a}}$hler reductions of the quaternion K$\ddot{\mathrm{a}}$hler Grassmannian $Gr_4(\mathbb{R}^8)$ by an isometric act…
On the Ricci curvature of Kahler-Ricci Flow
本文研究Kähler-Ricci流中Kodaira维数为一的流形的Ricci曲率收敛性，揭示其在局部拓扑下的收敛行为。
We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.
We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from K{ä}hler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metrics of the pseudo-Hermitian Einstein structures. These Fefferman metrics are never Einstein, but they are locally always conformally Einstein.
On the stability of Einstein metrics carrying a special twisted spinor
本文研究了携带特殊扭转旋量的爱因斯坦度量的稳定性，证明了其线性半稳定性并扩展了相关稳定性的结果。
A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-Kähler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time ${\cal N}$ is constructed whose extra dimensions corresp…
We obtain a necessary and sufficient condition of existence of a K{ä}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 not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge-Amp{è}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.
Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.
The projective algebra p(M;F) (i.e the collection of all projective vector fields)of a Finsler space (M;F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket. The projective algebra of Einstein metrics has been perpetually studied from physical and geometrical approaches. Here, the projective algebra of Einstein Matsumoto space of dimension n \geq 3 is characterized. Moreover, Einstein Matsumoto metrics with maximum projective symmetry are studied and characterized.
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
In this work an intrinsic projectively invariant distance is used to establish a new approach to the study of projective geometry in Finsler space. It is shown that the projectively invariant distance previously defined is a constant multiple of the Finsler distance in certain case. As a consequence, two projectively related complete Einstein Finsler spaces with constant negative scalar curvature are homothetic. Evidently, this will be true for Finsler spaces of constant flag curvature as well.
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate ap…
We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.
$\mathbb{T}^n$-invariant Kaehler-Einstein manifolds immersed in complex projective spaces
本研究给出n≤6时浸入带Fubini-Study度量的有限维复射影空间的非等距$\u27d4^n$-不变Kaehler-Einstein流形完整列表，解决了Calabi、Chern等关注的经典长期问题。
The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of the moduli spaces is independent of the Einstein-Hermitian constant, and rigidity of the associated real standard, and totally real maps is examined. Finally, certain classical results concerning embeddings of two-dimensional spheres into spheres are rephrased and derived in our…
We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurf…
Cohomogeneity one Einstein metrics on complex projective spaces
研究复射影空间上在紧连通李群余齐一作用下不变的爱因斯坦度量，假设奇异轨道为全测地，通过分析五个模型中的对角不变度量，证明其中四个模型不存在光滑整体定义的不变爱因斯坦度量，并给出剩余一个模型存在的必要条件。
Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four
本文研究四维情况下para-Kaehler度量的pc-仿射等价性，探讨其局部结构及Einstein类型的特征。
Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentzian Einstein metrics on simply connected manifolds, and describe all possible dimensions of the space of essential projective vector fields.
Two Kaehler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kaehler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected 2n-dimensional Riemannian Kaehler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we de…
We study the existence of weighted extremal Kähler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi. We obtain a ge…
An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a non-degenerate co-oriented hypersurface in flat affine space by its second fundamental form together with either the projective structure induced by the affine normal o…
We study the following problem: given an Einstein metric on a manifold, characterize and study all Einstein metrics which are pointwise projective to the given one. By definition, two metrics are said to be pointwise projectively related if they have the same geodesics as point sets. This is closely related to Hilbert's Fourth Problem. In this paper, we show that pointwise projectively related Einstein metrics satisfy a simple equation along geodesics. In particular, we prove the following rigidity theorem: if two pointwise projectively related Einstein metrics are complete with negative Einst…
We classify Kähler-Einstein manifolds which admit a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini-Study metric, whose codimention is not greater than 3 and whose metric is rotation invariant.
Families of projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics
本文构造了允许耦合凯勒-爱因斯坦度量但不允许可爱因斯坦度量的高维托里克粉丝 manifold，并提出在所有维度n≥4中此类例子存在的猜想。
We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective compactification of neutral signature Einstein metrics canonically defined on certain rank--$n$ affine bundles $M$ over $n$-dimensional manifolds endowed with projective structures.
On projective symmetries on Finsler Spaces
本文证明了Akbar-Zadeh的两种Einstein-Finsler空间定义在$I$-不变射影向量场积分曲线上等价，揭示了闭Finsler空间中射影对称性退化为Killing向量场的条件，并完成了紧致各向同性平均Landsberg流形的分类。
An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of Kähler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm $||f||_{#}$ along the or…
This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space.
In this short note, we investigate the existence of orbifold Kähler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric. Moreover, we characterize the K-stability of $\mathbb{Q}$-factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.
In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.
We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.
This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einstein metric.
Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also sufficient. We also obtain results for the problem of metrisability (without the Einstein condition): We show that the odd Chern type invariants of an affine connection are projective invariants that obstruct the existence of a projectively related Levi-Civita connection. In add…
In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.
Singular Kähler-Einstein metrics and RCD spaces
该研究聚焦奇异射影簇上的Kähler-Einstein度量，证明在常数量曲率度量的逼近条件下，其光滑部分的度量完备化为非坍塌RCD空间且与原簇同胚，拓展了奇异几何空间的研究范畴。
Kähler-Einstein toric submanifolds of the projective space
该研究证明了某些托卡蒂芬诺流形无法通过射影浸入诱导出凯勒-爱因斯坦度量，仅当其为射影线乘积时才可能实现，并提出了相关猜想。
Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and an analogous of Schwarz' lemma in Finsler geometry is proved. Next, the Schwarz' lemma is applied to show that the introduced pseudo-distance is a distance. This projectively invariant distance will be served in continuation of this work to investigate Einstein-Finsler spaces and classify Finsler spaces as well.
Inhomogeneous Einstein metrics on complex projective spaces
该研究针对复维数3≤n≤7的复射影空间，构造出首批非齐次爱因斯坦度量，对复偶维n=4、n=6的情形，肯定解答了伯杰关于是否存在非齐次爱因斯坦度量的问题。
Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
本文研究射影紧致Ricci平坦Einstein流形在类时/类空无穷远处的强Carrollian几何构造，通过引入基于非有效齐次模型的新型Cartan几何，揭示了其与底层射影结构的联系并证明其普遍性。
We focus on the classical open problem of the classification of Kähler-Einstein manifolds that can be Kähler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in the special case of Kähler- Einstein metrics admitting symmetries of rotational type. This leads to certain integrable distributions allowing a classification of such metrics.
We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [\nabla])$. We show that there is a canonical Einstein of metric on an $\R^*$--bundle over $T^*N$, with a connection whose curva…
We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano Kahler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.
A differential operator introduced by A. Gray on the unit sphere bundle of a Kähler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a Kähler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all Kähler-Einstein surfaces are then derived.
It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Ein…
In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.
Projectively induced Kähler--Einstein surfaces
本文分类了由全纯等距浸入复射影空间诱导度量的紧致Kähler--Einstein曲面，证明仅有射影平面和双曲面的乘积，且均为齐次。
Compatibility complexes for the conformal-to-Einstein operator
本文构建了 conformal-to-Einstein 运算符的兼容复形，揭示其最多只有一个独立解，并探讨了与局部平坦情况下的 Bernstein-Gelfand-Gelfand 序列的关系。
Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-ε}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce…
Kähler--Einstein metrics on quasi-projective manifolds
本文证明了在$K_X+D$为大且nef时奇异Kähler--Einstein度量的几乎完备性，并建立了仅大时负曲率锥形度量的弱收敛，部分回答了相关问题。
All invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces
本文完整分类了紧秩一对称空间切球丛上的所有不变切触度量结构，证明仅复射影空间情形存在3-Sasakian度量，并给出球面与实射影空间上唯一的Einstein切触度量，同时将这些结构延拓到穿孔切丛上的相应近Kähler、Kähler及hyperKähler结构。
We study the Kähler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_Σ\oplus L)$ over compact Kähler-Einstein manifold $Σ^n$. Assuming the initial Kähler metric $ω_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(Σ, L, [ω_0])$, under which the $\mathbb{P}^1$-fiber collapses along the Kähler-Ricci flow and the projective bundle converges to $Σ$ in Gromov-Hausdorff sense. Furthermore, the Kähler-Ricci flow must have Type I singularity and is of $(\C^n \times \mathbb{P}^1)$-type. This generalizes and extends part of Song-Weinkove's…
In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the variation of Bergman kernels and Kähler-Einstein volume form have been considered.
A characterization of complex projective space via the Calabi curvature operator
该研究针对Kähler-Einstein流形，利用含Calabi曲率算子的新Bochner公式证明了Tachibana型结果，分离出复射影空间，解决了AIM研讨会提出的相关问题。
Given a projective structure on a surface $N$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space $M$ of a certain rank $2$ affine bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on $\mathbb{RP}^2$ is the non-compact real form of the Fubi…
Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary.
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