The K?hler-Einstein metric for some Hartogs domains over symmetric domains

We study the complete K?hler-Einstein metric of a Hartogs domain built on an irreducible bounded symmetric domain sW, using a power N ^μ of the generic norm of Ω. The generating function of the K?hler-Einstein metric satisfies a complex Monge-Ampère equation with a boundary condition. The domain is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ε [0, 1[. This allows us to reduce the Monge-Ampère equation to an ordinary differential equation with a limit condition. This equation can be explicitly solved for a special value μ₀ of μ. We work out the details for the two exceptional symmetric domains. The special value μ₀ seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.     

The Khler-Einstein metric for some Hartogs domains over symmetric domains

We study the complete Kahler-Einstein metric of a Hartogs domainΩbuilt on an irreducible bounded symmetric domainΩ, using a power Nμof the generic norm ofΩ.The generating function of the Kahler-Einstein metric satisfies a complex Monge-Ampere equation with a boundary condition. The domainΩis in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X∈[0,1[. This allows us to reduce the Monge-Ampere equation to an ordinary differential equation with a limit condition. This equation can be explicitly solved for a special valueμ0 ofμ.. We work out the details for the two exceptional symmetric domains. The special valueμ0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.     

Existence of the Kähler-Einstein metric on certain Fano complete intersections

For certain families of Fano complete intersections we prove an estimate from below for the global (log) canonical threshold, which implies the existence of Kähler-Einstein metric on generic varieties in these families.     

Almost Kähler-Einstein Structures on 8-Dimensional Walker Manifolds

We study almost K?hler-Einstein structures on 8-dimensional Walker manifolds, i.e., pseudo-Riemannian 8-manifolds admitting a field of parallel null 4-planes, whence the metric is of neutral signature. We construct on explicit almost K?hler-Einstein Walker metrics which are not K?hler. An appropriate restriction induces examples of such metrics on the 8-torus, thereby producing a counterexample to Goldberg’s conjecture in the case of neutral signature. S. Haze passed away on 15 March 2006.     

The Bergman metric on complete Kähler manifolds

We use the existence of a bounded uniformly Hölder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kähler manifold. As an application, we proved that any simply-connected complete Kähler manifold with sectional curvature bounded above by a negative constant is Bergman complete. Mathematics Subject Classification (2000):32H10Supported by NSFC grant no. 10271089     

Kähler-Einstein metrics and projective embeddings

We prove that the complex projective space equipped with its Fubini-Study metric admits no compact Kähler-Einstein submanifold with nonpositive Einstein constant. In particular, the Calabi-Yau metrics carried by an algebraic K3 surface cannot be realized by projective embeddings.     

Weil-Petersson metric in the moduli space of compact polarized Kähler-Einstein manifolds of zero first Chern class

We study the problem of computing the curvature of the Weil-Petersson metric of the moduli space of general compact polarized Kähler-Einstein manifolds of zero first Chern class. We use canonical lifting of vector fields from the moduli space to the total deformation space to obtain a formula for the curvature of the Weil-Petersson metric. From this formula we obtain negative bisectional curvature for certain directions. This formula also reprove and explain the recent result of Schumacher that the holomorphic sectional curvature of the Weil-Petersson metric for K3-surfaces and symplectic manifolds are negative.     

Moving Symplectic Curves in Kähler-Einstein Surfaces

We derive a parabolic equation for the K?hler angle of a real surface evolving under the mean curvature flow in a K?hler-Einstein surface and show that a symplectic curve remains symplectic with the flow. Received March 29, 2000, Revised April 29, 2000, Accepted May 16, 2000     

Deformation of conic negative Kähler-Einstein manifolds

In this note, we investigate the behavior of a smooth flat family of n-dimensional conic negative Kähler-Einstein manifolds. By H. Guenancia’s argument, a cusp negative Kähler-Einstein metric is the limit of conic negative Kähler-Einstein metric when the cone angle tends to 0. Furthermore, it establishes the behavior of a smooth flat family of n-dimensional cusp negative Kähler-Einstein manifolds.     

Chern forms,chern numbers and curvature conditions on a Kähler-Einstein surface

In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL ²-norm of (3C ₂ ?C ₁ ² )/C ₁ ² is sufficiently small, whereC ₁ andC ₂ are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C ₂ ?C ₁ ² = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.     

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.     

The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.     

The Calabi metric for the space of Kähler metrics

Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation are given.     

On almost contact metric 1-hypersurfaces in Kählerian manifolds

We prove that an almost contact metric structure on an orientable hypersurface with type number 1 in a Kählerian manifold is necessarily cosymplectic.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~n

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n > 1) is an automorphism.     

K?hler–Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain ${M\subset X}$ in a complex manifold carries a complete K?hler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ${\partial M}$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over ${\partial M}$ . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K?hler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of K?hler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c ₁?<?0 admits a complete K?hler–Einstein metric.