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 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.
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.
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 $k \neq n/2$.
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.
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.
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.
On products of locally conformally Kähler manifolds
University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛瓦数学研究所）
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.
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.
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.
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.
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.
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 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.
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. $>$) $0$.
Let $M$ be a 5 dimensional Riemannian manifold with $Sec_M\in[0,1]$, $Σ$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists $\varepsilon_0>0$, such that $\int_Σ(1+H^2)^2 \ge 8π^2/3$, provided $H \le \varepsilon_0$. In particular, if $Σ$ is a locally conformally flat minimal hypersphere in $M$, then $Vol(Σ) \ge 8π^2/3$, which partially answer a question proposed by Mazet and Rosenberg \cite{Ma&Rosen}. For an $(n+1)-$ dimensional rotationally symmetric Riemannian manifold $M$, we show that an immersed hypersurface $Σ$ is locally conformally flat…
We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler metrics on Hopf manifolds, explicitly characterized by Vaisman. Finally, we review the properties of zero projectively flat metrics. As applications, we refine a list of possible projectively flat metrics by Li, Yau, and Zheng; moreover we prove that projectively flat astheno-Kähler…
We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is unimodular, while in the latter, we prove that if the Lie algebra is unimodular, then it is isomorphic to the product of $\mathbb R$ and a Heisenberg Lie algebra.
A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.
In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this …
We characterize compact locally conformally Kähler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. Then we characterize the Hopf manifolds, up to holomorphic isometry, as compact l.c.K. manifolds admitting a certain closed LCR action of $\mathbb{C}^*$.
We show that Oeljeklaus-Toma manifolds $X(K, U)$ where $K$ is a number field of signature $(s, t)$ such that $s\geq 1,$ $t\geq 2$ and $s\geq 2t$ admit no lck metric. Combined with the earlier results by K. Oeljeklaus - M. Toma and A. Dubickas this completely solves the problem of existence of LCK metrics on Oeljeklaus-Toma manifolds.
We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.
We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific G-structure of l.c.K. manifolds. We show that compact l.c.K. manifolds admitting a non-compact CC^* flow of LCR transformations are rigid: it is holomorphically conformal to a Hopf manifold with parallel Lee form.
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, w…
We show that a statistical manifold manifold of a constant non-zero curvature can be realised as a level line of Hessian potential on a Hessian cone. We construct a Sasakian structure on $TM\times\R$ by a statistical manifold manifold of a constant non-zero curvature on $M$. By a statistical Lie algebra of a constant non-zero Lie algebra we construct a l.c.K Lie algebra.
We study the equation $Δ_g u -\frac{n-2}{4(n-1)}R(g)u+Ku^p=0 (1+ζ\leq p \leq \frac{n+2}{n-2})$ on locally conformally flat compact manifolds $(M^n,g)$. We prove the following: (i) When the scalar curvature $R(g)>0$ and the dimension $n \geq 4$, under suitable conditions on $K$, all positive solutions $u$ have uniform upper and lower bounds; (ii) When the scalar curvature $R(g)\equiv 0$ and $n \geq 5$, under suitable conditions on $K$, all positive solutions $u$ with bounded energy have uniform upper and lower bounds. We also give an example to show that the energy bound condition for the unifo…
A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of conformal classes on compact manifolds containing two non-homothetic Kähler metrics. The second one is the classification of compact Einstein locally conformally Kähler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compa…
