The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

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.     

On Hamiltonian stationary Lagrangian spheres in non-Einstein K?hler surfaces

Hamiltonian stationary Lagrangian spheres in K?hler-Einstein surfaces are minimal. We prove that in the family of non-Einstein K?hler surfaces given by the product Σ₁?×?Σ₂ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces Σ₁ and Σ₂ are spheres, is unstable.     

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.     

Homogeneous nearly K?hler manifolds

The structure of nearly K?hler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233?C248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly K?hler manifold is locally a Riemannian product of homogeneous nearly K?hler spaces, twistor spaces over quaternionic K?hler manifolds and six-dimensional (6D) nearly K?hler manifolds, where the homogeneous nearly K?hler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly K?hler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K?hler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.     

Killing spinors on K?hler manifolds

In the paper Kählerian Killing spinors are defined and their basic properties are investigated. Each Kähler manifold that admits a Kählerian Killing spinor is Einstein of odd complex dimension. Kählerian Killing spinors are a special kind of Kählerian twistor spinors. Real Kählerian Killing spinors appear for example, on closed Kähler manifolds with the smallest possible first eigenvalue of the Dirac operator. For the complex projective spaces P ^2l–1 and the complex hyperbolic spaces H ^2l–1 withl>1 the dimension of the space of Kählerian Killing spinors is equal to ( ). It is shown that in complex dimension 3 the complex hyperbolic space H ³ is the only simple connected complete spin Kähler manifold admitting an imaginary Kählerian Killing spinor.     

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.     

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.     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

Local Schr?dinger flow into K?hler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schr?dinger flow for maps from a compact Riemannian manifold into a complete K?hler manifold, or from a Euclidean space R^m into a compact K?hler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.     

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.      

Finite morphisms of projective and K?hler manifolds

In this paper we study finite morphisms of projective and compact K?hler manifolds, in particular, positivity properties of the associated vector bundle, deformation theory and ramified endomorphisms. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday     

Kähler manifolds with numerically effective Ricci class and maximal first Betti number are tori

Let M be a n-dimensional Kähler manifold with numerically effective Ricci class $c_1(M)$. In this Note we prove that, if the first Betti number $b_1(M)$ is equal to 2n, then M is biholomorphic to a n-dimensional complex torus. To cite this article: F. Fang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Hodge decomposition theorem on strongly K?hler Finsler manifolds

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the -Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly K?hler Finsler manifold, we define a -Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in K?hler manifolds is still true in the more general compact strongly K?hler Finsler manifolds. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

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.     

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.