Kähler–Einstein metrics along the smooth continuity method

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kähler–Einstein manifolds. We also give analogous results for twisted Kähler–Einstein metrics and Kahler–Ricci solitons.     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

The Kähler–Ricci flow with positive bisectional curvature

We show that the Kähler–Ricci flow on a manifold with positive first Chern class converges to a Kähler–Einstein metric assuming positive bisectional curvature and certain stability conditions.     

On Ricci solitons of cohomogeneity one

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler–Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.     

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.     

A uniqueness theorem in Kähler geometry

We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F(Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler–Einstein.     

Anticanonically balanced metrics on Fano manifolds

Annals of Global Analysis and Geometry - We show that if a Fano manifold has discrete automorphism group and admits a polarized Kähler–Einstein metric, then there exists a sequence of...     

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kähler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kähler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kähler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kähler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kähler with either a closed fundamental four form or fundamental two forms defining a differential ideal.     

Stability of Einstein-Hermitian vector bundles

Einstein-Hermitian vector bundles are defined by a certain curvature condition. We prove that over a compact Kähler manifold a bundle satisfying this condition is semistable in the sense of Mumford-Takemoto and a direct sum of stable Einstein-Hermitian subbundles.     

Rotationally symmetric extremal pseudo-Kähler metrics of non-constant scalar curvatures

In this paper, we explicitly construct some rotationally symmetric extremal (pseudo-)Kähler metrics of non-constant scalar curvature, which depend on some parameters, and on some line bundles over projective spaces. We also discuss the phase change phenomenon caused by the variation of parameters.     

K‐Stability and Kähler‐Einstein Metrics

We prove that if a Fano manifold M is K‐stable, then it admits a Kähler‐Einstein metric. It affirms a longstanding conjecture for Fano manifolds. © 2015 Wiley Periodicals, Inc.     

On transversally symmetric pseudo-Einstein and Fefferman–Einstein spaces

We describe and construct pseudo-Hermitian structures θ without torsion (i.e. with transverse symmetry) whose Webster–Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures TS-pseudo-Einstein and our first result states that all these structures can locally be derived from Kähler–Einstein metrics. Then we discuss the corresponding Fefferman metrics of the TS-pseudo-Einstein structures. These are never Einstein. However, our second result states that they are locally always conformally Einstein.     

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.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

Existence of a complete holomorphic vector field via the Kähler–Einstein metric

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.