We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including Kahler-Einstein metrics and conformally Einstein Kahler metrics. We also extend some of our results to almost Kahler 4-manifolds, providing new examples of Ricci-flat almost Kahler metrics which are not Kahler.
We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole eq…
I present a construction of real or complex selfdual conformal 4-manifolds (of signature (2,2) in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex 2-manifold. The 4-manifolds obtained are characterized by the existence of a foliation by selfdual null surfaces of a special kind. The classification by Dunajski and West of selfdual conformal 4-manifolds with a null conformal vector field is the special case in which the gauge group reduces to the group of diffeomorphisms commuting wi…
It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain `…
Any strictly pseudoconvex domain in C2 carries a complete Kahler-Einstein metric, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kahler-Eins…
Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compa…
On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a Kähler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the Kähler metric admits a special Kähler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the Kähler metric is part of an ambitoric structure, in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toric one. The formulas are derived using the Duisterm…
We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series. It follows that WBF Kahler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kahler 6-manifolds, extending work by the first three authors on weakly selfdual Kahler 4-manifolds. The const…
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishe…
We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge1$ pseudoconvex boundary components and at least one non-parabolic end, the first betti number of $K$ is at least $l$. In either case, any boundary component has non-vanishing first betti number. If $K$ has one pseudoconvex boundary component with vanishing first betti…
This paper has been withdrawn in order to replace it by two separate submissions: 1. Hamiltonian 2-forms in Kahler geometry III: Extremal metrics and stability, math.DG/0511118; 2. Hamiltonian 2-forms in Kahler geometry IV: Weakly Bochner-flat Kahler manifolds, math.DG/0511119. As the titles indicate, the first paper covers the parts of this withdrawn submission concerning extremal Kahler metrics, while the second one deals the weakly Bochner-flat Kahler metrics. However, the material for the first paper has been substantially revised and extended with several new results. 1. The exposition ha…
In complex Finsler geometry, an open problem is: does there exist a weakly Kähler Finsler metric which is not Kähler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly Kähler Finsler metrics which are non-Kähler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly Kähler condition.
Motivated by the torus stability problem, in this work we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly.
We study the complete Kahler-Einstein metric in tube domains. We obtain estimates of this metric and its holomorphic bisectional curvatures near the weakly pseudoconvex boundary points.
Let $M_1$ and $M_2$ be two Kähler manifolds. We call $M_1$ and $M_2$ {\em relatives} if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S\to M_1$ and $h_2: S\to M_2$. Moreover, two Kähler manifolds $M_1$ and $M_2$ are said to be {\em weakly relatives} if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds $S_1$ and $S_2$ which admit two Kähler immersions into $M_1$ and $M_2$ respectively. The notions introduced are not equivalent. Our main results in this paper are Theorem 1.…
We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective…
We present a classification of compact Kaehler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kaehler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kaehler manifolds with a rigid hamiltonian torus action are bundles of toric Kaehler manifolds. The second…
Let $ l : Σ\to X$ be a weakly Lagrangian map of a compact orientable surface $ Σ$ in a Kähler surface $ X$ which is area minimizing in its homotopy class of maps in $ W^{1,2}(Σ, X)$, the Sobolev space of maps of square integrable first derivative. Schoen and Wolfson showed such $ l$ is Lipschitz, and it is smooth except at most at finitely many points of Maslov index 1 or -1. In this note, we observe if in addition c_1(X)[l]=0, $ l$ is smooth everywhere. Here $ c_1(X)$ is the first Chern class of $ X$.
