Liviu Ornea
I present a selection of results on locally conformally Kähler geometry published after 1997. The proofs are mainly sketched, some of them are even omitted. Several open problems are indicated in the end.
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We show that every Kato surface (or surface with a global spherical shell) admits a locally conformally Kaehler metric.
A moment map for twisted-Hamiltonian vector fields on locally conformally Kähler manifolds
本文将Kähler几何中标量曲率作为矩映射的经典Donaldson-Fujiki解释推广至局部共形Kähler几何框架。
We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.
The paper is part of an attempt of understanding non-Kähler threefolds. We start by looking at compact complex non-Kähler threefolds with algebraic dimension two and admitting locally conformally Kähler metrics. Under certain assumptions, we prove that they are blown-up quasi-bundles over a projective surface.
Locally conformally Kähler spaces and proper open morphisms
本文研究了带奇点的局部共形Kähler空间的稳定性，找到了全纯映射下像空间仍容许lcK度量的充分条件，推广了Varouchas关于Kähler空间的结果。
In this paper, we study the properties of coverings of locally conformally Kähler (LCK) spaces with singularities. We begin by proving that a space is LCK if any only if its universal cover is Kähler, thereby generalizing a result from a previous paper. We then show that a complex space which projects over an LCK space with discrete fibers must also carry an LCK structure.
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。
A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic potential. This version was added 10 years after publication to correct some errors in the original.
A characterization of Oeljeklaus-Toma manifolds in locally conformally Kähler geometry
本文刻画一类可解 Lie 群，证明其若具有左不变非 Vaisman 局部共形 Kähler 度量和格，则必来自 Oeljeklaus-Toma 构造，从而解释数论在其中的作用。
We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we make a complete classification of 4-dimensional homogeneous and locally homogeneous l.c.K. manifolds in terms of Lie algebras.
We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to locally conformally balanced manifolds by Hilbert-Chow map.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kaehler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus, and a metric structure theorem asserting that it is necessarily of Vaisman type. We also discuss and determine l.c.K. reductive Lie groups and compact locally homogeneous l.c.K. manifolds of reductive Lie groups.
We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is locally conformally Kaehler, and, in fact, Vaisman. This leads to classification results for normal metric contact pairs. In complex dimension two we obtain a new proof of Belgun's classification of Vaisman manifolds under the additional assumption that the Kodaira dimension is non-n…
We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension $4k$ in terms of integrality of eta invariants.
In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.
Bi-Hermitian and locally conformally Kähler surfaces
本文探讨复曲面上双 Hermite 度量与局部共形 Kähler 度量之间的若干相互关系。
We give a short proof of the fact that compact pluricanonical locally conformally Kähler manifolds have parallel Lee form.
We characterize the existence of a locally conformally Kähler metric on a compact complex manifold in terms of currents, adapting the celebrated result of Harvey and Lawson for Kähler metrics.
We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.
We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of noncompact normalizer and determine all left-invariant locally conformally Kaehler structures on reductive Lie groups.
We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic $2$-form which tames the underlying almost complex structure.
We prove that locally conformally Kähler metrics on certain compact complex surfaces with odd first Betti number can be deformed to new examples of bi-Hermitian metrics.
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.
The existence of negatively curved metrics on locally conformally flat manifolds with boundary
本文研究带边界流形上共形度量的存在性，借助 Morse 函数构造负截面曲率度量，并在无共形平坦性假设时构造正 Einstein 张量度量。
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.
It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.
Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
In this paper we generalize the basic Lichnerowicz cohomology on transversally locally conformally Kählerian foliations and we study its relation with basic Bott-Chern cohomology and $0$--th basic Dolbeault cohomology with values in the associated foliated weight bundle.
We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.
Non-linear Hopf manifolds are locally conformally Kahler
本文证明由非线性全纯收缩映射定义的 Hopf 流形可全纯嵌入线性 Hopf 流形，并因此具有局部共形 Kähler 度量。
We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also show that any eternal solutions to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space-time maximum must be a steady Yamabe soliton.
A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.
A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.
Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifolds. As applications of these results we exhibit explicit examples of metrics g, conformal to g they are solutions for this problem.
It is known that automorphism group $G$ of a compact homogeneous locally conformally Kähler manifold $M=G/H$ has at least a 1-dimensional center. We prove that the center of $G$ is at most 2-dimensional, and that if its dimension is 2, then $M$ is Vaisman and isometric to a mapping torus of an isometry of a homogeneous Sasakian manifold.
Structure of locally conformally symplectic Lie algebras and solvmanifolds
该研究针对局部共形辛李代数展开结构分析，完成了四维李代数上局部共形辛结构的分类，还在四维连通单连通可解李群的紧商上构造了这类结构。
On products of locally conformally Kähler manifolds
本文证明了正维紧复流形的乘积不存在严格局部共形Kähler度量，解决了lcK几何中长期未解的猜想。
In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.
We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.
Cohn-Vossen theory for locally conformally flat manifolds
通过建立非负n-超调和函数的精细奇异性估计，解析刻画了非负Ricci曲率完备非紧局部共形平坦流形的体积增长，验证了Yau关于Cohn-Vossen不等式的猜想，并证明了消除早期工作所有辅助假设的尖锐间隙定理。
We give an equivalent definition of compact locally conformally hyperkähler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that a compact Kähler manifold admitting a holomorphic symplectic form is hyperkähler.
In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.
We study the condition in which G2-structures are introduced by a non closed four-form, although they are satisfying locally conformal conditions.All solutions are found in the case when the Lee form of G2-structures is non-zero and gintroduces seven-dimensional Lie algebras, The main results are given in preposition1 and theorem1.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than $\frac {n-α}2$ for some $α\in [2, n)$, the set of positive smooth solutions to the equation $$ P_αu = u^\frac {n+α}{n-α} $$ is compact in the $C^\infty$ topology. There…
Locally conformally homogeneous Lorentzian spaces
该研究证明了局部共形同构的洛伦兹流形要么共形平坦，要么局部等价于齐性平面波，并揭示了平面波结构与Penrose极限的关系。
Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to be globally defined. We investigate both the geometry of locally conformally cosymplectic (abbreviated as LCC) manifolds and the Hamiltonian dynamics constructed on such LCC manifolds. Further, we provide a geometric Hamilton-Jacobi theory on this geometric framework.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kahler metrics with potential which could arise from an LCK structure is in bijective correspondence with the s…
We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the $4$-dimensional structures in our classification.
We present an overview of recent results in locally conformally Kähler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.
We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is closed. We further prove a de Rham type splitting theorem which states that if such a manifold is analytic, it is isometric to the Riemannian product of a Euclidean space and an incomplete manifold.
This is a survey article on the existence of locally conformally flat(LCF) and self-dual(SD) metrics on various basic 4-manifolds like simply-connected ones or product types
Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,θ)$ a $d$(bounded) locally conformally Kähler manifold if the lifted Lee form $\tildeθ$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}χ(M^{2n})\geq$ (resp. $>$) $0$.
In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^\infty$ topology.
HKT geometry on locally conformally hyperkähler manifolds
研究紧致局部共形超凯勒流形上的HKT几何，证明四元数蒙日-安培方程可解，确认HKT-Ricci流最大存在时间猜想，并在四元数Hopf曲面上构造非平凡HKT孤子及证明归一化流收敛。
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