Deformations of Almost-Kähler Metrics with Constant Scalar Curvature on Compact Kähler Manifolds

We prove the existence of infinite-dimensional families of(non-Kähler) almost-Kähler metrics with constant scalar curvature oncertain compact manifolds. These are obtained by deformingconstant-scalar-curvature Kähler metrics on suitable compact complexmanifolds. We prove several other similar results concerning the scalarcurvature and/or the *-scalar curvature. We also discuss thescalar curvature functions of almost-Kähler metrics.     

Extremal Kähler metrics and complex deformation theory

Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [] has an open neighborhood inH ^1,1 (M, ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [] is nondegenerate, every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.The first author is supported in part by NSF grant DMS 92-04093.     

On the existence and nonexistence of extremal metrics on toric Kähler surfaces

In this paper we study the existence of extremal metrics on toric Kähler surfaces. We show that on every toric Kähler surface, there exists a Kähler class in which the surface admits an extremal metric of Calabi. We found a toric Kähler surface of 9 -fixed points which admits an unstable Kähler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of toric surfaces by simple piecewise linear functions. As an application, we show that among all toric Kähler surfaces with 5 or 6 -fixed points, is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

The Abreu equation with degenerated boundary conditions

The Abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds. We study the Dirichlet problem of the Abreu equation with degenerated boundary conditions. The solutions provide the Kähler metrics of constant scalar curvature on the complex torus.     

On the K-stability of complete intersections in polarized manifolds

We consider the problem of existence of constant scalar curvature Kähler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai–Umemura–Tian like example of Fano 5-fold admitting no Kähler–Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.     

环流形上的极值度量——存在性和K-稳定性

本文主要研究环流形上的极值度量的存在性和K-稳定性.本文将Donaldson关于环流形上有关常数量曲率度量的稳定性概念的约化推广到一般的极值度量的情形.通过这个约化,本文证明环流形上极值度量的存在性可以推出流形对于环形变的相对K-稳定性.在不知道是否存在极值度量的情形下,本文还给出环流形相对K-稳定的一个充分性条件.对环曲面的情形,基于Arrezo-Pacard-Singer的工作,本文证明任意一个环曲面上存在含有极值度量的Ka¨hler类,并给出一些环曲面上有不存在极值度量的K¨ahler类的例子.关于一般的环流形上的极值度量的存在性,本文用变分方法研究其弱解,证明在能量泛函逆紧性假设下,存在弱极小化子.     

Four-Dimensional Almost Kähler Einstein and *-Einstein Manifolds

In the framework of studying the integrability of almost Kähler manifolds, we prove that a four-dimensional almost Kähler Einstein and -Einstein manifold is a Kähler manifold. Further, we estimate the *-scalar curvature of a four-dimensional compact almost Kähler Einstein and weakly *-Einstein manifold with negative scalar curvature.     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Minimal two spheres in Kähler-Einstein Fano manifolds

In this paper we show the existence of stable symplectic non-holomorphic two-spheres in Kähler manifolds of positive constant scalar curvature of real dimension four and in Kähler-Einstein Fano manifolds of real dimension six. Some of the techniques used involve deformation theory of algebraic cycles.     

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.     

Positive Quaternion Kähler Manifolds with fourth Betti number equal to one

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. In this article we are mainly concerned with Positive Quaternion Kähler Manifolds M satisfying b₄(M)=1. Generalising a result of Galicki and Salamon we prove that M4n in this case is homothetic to a quaternionic projective space if 2≠n?6.     

Blowing up and desingularizing constant scalar curvature Kähler manifolds

This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.     

A Remark on Kahler Metrics of Constant Scalar Curvature on Ruled Complex Surfaces

In this paper we point out how some recent developments in thetheory of constant scalar curvature Kähler metrics canbe used to clarify the existence issue for such metrics in thespecial case of (geometrically) ruled complex surfaces. 2000Mathematics Subject Classification 53C55, 58E11.     

The Geometry of Positive Locally Quaternion Kähler Manifolds

A classification of locally quaternion Kähler manifolds M ⁴ⁿ with positive scalar curvature is obtained as a consequence of J. Wolf's work on space forms of irreducible symmetric spaces. We determine the Betti numbers of such manifolds M ⁴ⁿ as well as of the projective 3-Sasakian manifolds fibering over them. We study the geometry of the quaternion Kähler and locally quaternion Kähler submanifolds for each M ⁴ⁿ, which is particularly significant for 4n = 16 due to its relation with four quaternionic structures on the Grassmannian (R ⁸).     

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds

In this paper,we discuss a Kazdan-Warner typed equation on certain non-compact Rie- mannian manifolds.As an application,we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundles over certain non-compact K(?)hler manifolds.     

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

     

Hyper-Hermitian Quaternionic Kähler Manifolds

We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space.     

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g ₀. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.     

The Calabi flow on Kähler Surfaces with bounded Sobolev constant (I)

We consider the formation of singularities along the Calabi flow by assuming the uniformly bounded Sobolev constants. On Kähler surfaces we prove that if curvature tensor is not uniformly bounded, then one can form a singular model called deepest bubble; such deepest bubble has to be a scalar flat ALE Kähler metric. In certain Kähler classes on toric Fano surfaces, the Sobolev constants are a priori bounded along the Calabi flow with small Calabi energy. We can also show in certain cases no deepest bubble can form along the flow. It follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently. To illustrate our results more clearly, we focus on an example on \({\mathbb{CP}^2}\) blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on \({\mathbb{CP}^2}\) blown up three points at generic position in the Kähler classes where the exceptional divisors have the same area.