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.     

On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kähler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kähler potentials on which this inequality holds on higher-dimensional Fano Kähler-Einstein manifolds.     

On the Kähler geometry of the Hilbert-Schmidt Grassmannian

We compute the Riemann curvature tensor of the Hilbert-Schmidt Grassmannian with respect to its natural Kähler structure. The sectional curvature is shown to be non-negative. We also discuss the Kähler structure of the Hilbert-Schmidt space of almost complex structures whose sectional curvature is shown to be non-positive.Research supported by a grant from the Osk. Huttunen Foundation and the Wihuri Foundation (Finland).     

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.     

Rozansky–Witten invariants via Atiyah classes

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of weights (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.     

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.     

Curvature of invariant metrics on Pyatetskii-Shapiro's generalized homogeneous domains

We study the curvature of invariant metrics on the generalization of the classical homogeneous domain of Pyatetskii-Shapiro, as given by D'Atri in [3]. We obtain all invariant Kähler metrics of either, nonpositive sectional curvature or nonpositive holomorphic sectional curvature, and determine the corresponding connected groups of isometries in each case. This yields a continuous family of nonsymmetric homogeneous Kähler metrics with nonpositive curvature.Supported in part by CONICOR and SECyT (UNC).     

Curvature properties of twistor spaces of quaternionic Kähler manifolds

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.Dedicated to Professor Helmut Karzel on the occasion of his 70th birthdayResearch partially supported by Contracts MM 413/1994 and MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.     

Holomorphic Spinors and the Dirac Equation

In the first part of this paper, the closed spin Kähler manifolds of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator, are characterized by holomorphic spinors. In the second part, the space of holomorphic spinors on a Kähler–Einstein manifold is described by eigenspinors of the square of the Dirac operator and vanishing theorems for holomorphic spinors are proved.     

Local Rigidity of Certain Classes of Almost Kähler 4-Manifolds

We show that any non-Kähler, almost Kähler 4-manifoldfor which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kähler metric is locally isometric to the (only)proper 3-symmetric four-dimensional space.     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

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.     

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kähler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||||² is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K.     

Four-dimensional almost Kähler Einstein manifolds

Summary We prove that a four-dimensional compact almost Kähler manifold which is Einsteinian, and Einsteinian is a Kähler manifold.     

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.     

Nearly Kähler manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

We study the local structures of nearly Kähler manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke (TV). We show that there does not exist a TV Bochner flat strict nearly Kähler manifold in 2n(?10) dimension and determine the local structures of the manifolds in 6 and 8 dimensions.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.