Compactness Theorems of Extremal-Kähler Manifolds with Positive First Chern Class

In this article, we propose an extension of the compactness property for Kähler–Einstein metrics to extremal-Kähler metrics on compact Kähler manifolds with positive first Chern class and admitting non-zero holomorphic vector fields.     

Constructions of Kähler–Einstein metrics with negative scalar curvature

We show that on Kähler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by Tsuji converges uniformly to the Kähler–Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji’s iterative construction.     

Métriques autoduales sur la boule

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 vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-K?hler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-K?hler metric on the (4m)-ball. Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002     

Rigidity of quasi-Einstein metrics

We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kähler quasi-Einstein metrics.     

Symplectic obstructions to the existence of ω-compatible Einstein metrics

It is shown that the existence of an ω-compatible Einstein metric on a compact symplectic manifold (M,ω) imposes certain restrictions on the symplectic Chern numbers. Examples of symplectic manifolds which do not satisfy these restrictions are given. The results offer partial support to a conjecture of Goldberg.     

Einstein condition and twistor spaces of compatible partially complex structures

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric. A generalization of the Einstein condition (discussed in the Besse book [Enstein Manifolds, Ergeb. Math. Grensgeb. (3), vol. 10, Springer, New York, 1987]) is also considered.     

Computing Perelman?s ν-functional

This is a short note in which we show how to calculate the value of Perelman?s ν-functional for a variety of metrics. In particular we complete the calculation of values for the known 4-dimensional Einstein and shrinking Ricci soliton metrics.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

The Sasaki Join,Hamiltonian 2-Forms,and Constant Scalar Curvature

We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki–Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension \(2n+1\) and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds \(M_{l_1,l_2,\mathbf{w}}\) of dimension \(2n+3\) which depend on four integer parameters \(l_1,l_2,w_1,w_2\). Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two-dimensional subcone of Sasaki–Ricci solitons in the Sasaki cone, and a unique Sasaki–Einstein metric in each of the two-dimensional subcones.     

Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds

This paper studies several aspects of asymptotically hyperbolic (AH) Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness.     

Critical metrics of the total scalar curvature functional on 4‐manifolds

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980's that every CPE metric must be Einstein. We prove that a 4‐dimensional CPE metric with harmonic tensor must be isometric to a round sphere      

Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the(1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics.More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are K¨ahler Calabi–Yau surfaces and Hopf surfaces.     

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.     

Curvature tensors of singular spaces

A sequence of tensor-valued measures of certain singular spaces (e.g., subanalytic or convex sets) is constructed. The first three terms can be interpreted as scalar curvature, Einstein tensor and (modified) Riemann tensor. It is shown that these measures are independent of the ambient space, i.e., they are intrinsic. In contrast to this, there exists no intrinsic tensor-valued measure corresponding to the Ricci tensor.     

Kähler-Ricci solitons on toric manifolds with positive first Chern class

In this paper we prove there exists a Kähler-Ricci soliton, unique up to holomorphic automorphisms, on any toric Kähler manifold with positive first Chern class, and the Kähler-Ricci soliton is a Kähler-Einstein metric if and only if the Futaki invariant vanishes.     

Generalised Einstein Equations and Prescribed Relations for the First Chern Weil Form

Generalised Einstein equations (Einstein equations with sources in thephysicist's grammar) can, in the Kähler setup, be seen ascohomological equations within the first Chern class. Introducing a twoparameter secondary class (or source term) to prescribe such acohomological relation, we characterise regions and paths of thoseparameters to ensure that the associated equation admits at least onesolution. Those regions could be seen, in the context of geometricanalysis, as a measure of the metric flexibility allowed within theKählerian rigidity. When the first Chern class is positive, the AubinTian constant and the bounds for the pluriharmonic concavity andconvexity of the source term characterise the bounds of that region.Taking into account the minimal regularity of the secondary class toensure the existence of classical solutions, we observe, in particular,an improvement of some results quoted in the literature in the contextof Calabi's conjecture.     

The Ricci tensor of almost parahermitian manifolds

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi–Civita connection. The formula uses the intrinsic torsion of an underlying \(\mathrm {SL}(n,\mathbb {R})\)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakähler version of the Goldberg conjecture and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics.