Interpolation in non-positively curved Kähler manifolds

We extend to any complete simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in\(\mathbb{C}\) and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tools are L² estimates and a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu, and Siu, Yau.     

Immersion of Kähler manifolds in the class of convex submanifolds

It is proved that a real complete convex Kähler submanifold in Euclidean space splits as a metric product of two-dimensional surfaces of positive Gaussian curvature in Euclidean 3-space and a Euclidean subspace. A theorem of V. K. Beloshapka and S. N. Bychkov is generalized to the case of convex submanifolds of any codimension.     

On the Gromov width of homogeneous Kähler manifolds

We compute the Gromov width of homogeneous Kähler manifolds with second Betti number equal to one. Our result is based on the recent preprint [4] and on the upper bound of the Gromov width for such manifolds obtained in [6].     

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.     

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.     

Pseudo-holomorphic curves in nearly Kähler manifolds

We study pseudo-holomorphic curves in general nearly Kähler manifolds. For that purpose, we first introduce the fundamental equations of submanifold geometry in terms of the characteristic connection of the nearly Kähler structure. Then we classify pseudo-holomorphic curves with parallel second fundamental form in Chern-flat nearly Kähler manifolds. Moreover, we give a new Simons type identity. As an application of this identity, we show that the closed pseudo-holomorphic curves in Chern-flat nearly Kähler manifolds with a second fundamental form of controlled growth are totally geodesic.     

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 .     

Equivariant Kähler compactifications of homogeneous manifolds

A compact complex manifoldX is an equivariant compactification of a homogeneous manifoldG/H (G a connected complex Lie group,H a closed complex subgroup ofG), if there exists a holomorphic action ofG onX such that theG-orbit of some pointx inX is open and H is the isotropy group ofx. GivenG andH, for some groups (e.g.,G nilpotent) there are necessary and sufficient conditions for the existence of an equivariant Kähler compactification which are proven in this paper.     

Transformations of locally conformally Kähler manifolds

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.     

On quasi-Sasakian hypersurfaces of Kähler manifolds

We establish several conditions which are necessary for a quasi-Sasakian hypersurface of a Kähler manifold to be minimal.     

The energy of a Kähler class on admissible manifolds

On a compact complex manifold of Kähler type, the energy E(Ω) of a Kähler class Ω is given by the squared L ²-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kähler metric representing the said class, and any one such metric whose scalar curvature has squared L ²-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kähler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kähler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kähler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.     

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.      

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.     

On the groups of c-projective transformations of complete Kähler manifolds

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini–Study metric. This establishes a stronger version of the recently proved Yano–Obata conjecture for complete Kähler manifolds.     

Hamiltonian stability of Lagrangian tori in toric Kähler manifolds

Let (M,J,ω) be a compact toric Kähler manifold of dim_? M=n and L a regular orbit of the T ⁿ-action on M. In the present paper, we investigate Hamiltonian stability of L, which was introduced by Y.-G. Oh (Invent. Math. 101, 501–519 (1990); Math. Z. 212, 175–192) (1993)). As a result, we prove any regular orbit is Hamiltonian stable when (M,ω)=??ⁿ,ω_FS) and (M,ω)=??ⁿ ₁× ??ⁿ ₂,aω_FS⊕ bω_FS), where ω_FS is the Fubini–Study Kähler form and a and b are positive constants. Moreover, they are locally Hamiltonian volume minimizing Lagrangian submanifolds.