Locally conformal Kähler manifolds with potential

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold ${\widetilde M}A locally conformally K?hler (LCK) manifold M is one which is covered by a K?hler manifold [(M)\tilde]{\widetilde M} with the deck transformation group acting conformally on [(M)\tilde]{\widetilde M}. If M admits a holomorphic flow, acting on [(M)\tilde]{\widetilde M} conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).     

Kähler manifolds with quasi-constant holomorphic curvature

The aim of this article is to classify compact Kähler manifolds with quasi-constant holomorphic sectional curvature.     

Kähler manifolds with homothetic foliation by curves

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliations by curves.     

On some Kähler manifolds with Norden metric

We investigate the Kähler manifolds with Norden metric whose curvature tensor can be expressed in the terms of Ricci tensors only and whose holomorphically conformal curvature tensor vanishes.     

Compact Kähler manifolds with nonpositive bisectional curvature

Let (M ⁿ , g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N ^k with c ₁ <  0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.     

Isospectral nearly Kähler manifolds

We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\). We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions for the investigated manifolds to be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifolds of the considered type is determined.     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR _IV(2)) to any Riemannian manifold with finite energy has to be constant.     

Homogeneous Kähler and Hamiltonian manifolds

We consider actions of reductive complex Lie groups \({G=K^\mathbb{C}}\) on Kähler manifolds X such that the K-action is Hamiltonian and prove then that the closures of the G-orbits are complex-analytic in X. This is used to characterize reductive homogeneous Kähler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit K-moment maps if and only if their isotropy groups are algebraic.     

Kähler magnetic fields on Kähler manifolds of negative curvature

On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.     

Locally conformally flat Kähler and para-Kähler manifolds

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kähler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP ^m or Gr2(C ^m+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.     

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

In a given Kähler manifold (M,J) we introduce the notion of Kähler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kähler immersions of M into an ambient Kähler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .