Examples of almost Einstein structures on products and in cohomogeneity one

Almost Einstein manifolds are conformally Einstein up to a scale singularity, in general. This notion comes from conformal tractor calculus. In the current paper we discuss almost Einstein structures on closed Riemannian product manifolds and on 4-manifolds of cohomogeneity one. Explicit solutions are found by solving ordinary differential equations. In particular, we construct three families of closed 4-manifolds with almost Einstein structure corresponding to the boundary data of certain unimodular Lie groups. Two of these families are Bach-flat, but neither (globally) conformally Einstein nor half conformally flat. On products with a 2-sphere we find an exotic family of almost Einstein structures with hypersurface singularity as well.     

Einstein Metrics with Prescribed Conformal Infinity on 4-Manifolds

This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions. Received: April 2006, Revision: November 2006, Accepted: February 2008     

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics

In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author in (Bahuaud, Pac. J. Math. 239(2): 231–249, 2009), we prove that decay of sectional curvature to ?1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result.     

A generalization of a 4-dimensional Einstein manifold

A weakly Einstein manifold is a natural generalization of a 4-dimensional Einstein manifold. In this paper, we shall give a characterization of a weakly Einstein manifold in terms of so-called generalized Singer-Thorpe bases. As an application, we prove a generalization of the Hitchin inequality for compact weakly Einstein 4-manifolds. Examples are provided to illustrate the theorems.     

Compactification of a class of conformally flat 4-manifold

In this paper we generalize Huber’s result on complete surfaces of finite total curvature. For complete locally conformally flat 4-manifolds of positive scalar curvature with Q curvature integrable, where Q is a variant of the Chern-Gauss-Bonnet integrand; we first derive the Cohn-Vossen inequality. We then establish finiteness of the topology. This allows us to provide conformal compactification of such manifolds. Oblatum 3-III-1999 & 18-II-2000?Published online: 8 May 2000     

Ricci flow of conformally compact metrics

In this paper we prove that given a smoothly conformally compact asymptotically hyperbolic metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact and asymptotically hyperbolic. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.     

Asymptotic Simplicity and Static Data

The present article considers time-symmetric initial data sets for the vacuum Einstein field equations, which are conformally related to static initial data sets in such a way that in a neighbourhood of infinity the two initial data sets have the same massless part. It is shown that for this class of data, the solutions to the regular finite initial value problem at spatial infinity for the conformal Einstein field equations extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.     

Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with specified asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.     

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi–Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi–Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang–Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian–Einstein connections on the Calabi–Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian–Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.     

U(1)-decomposable self-dual manifolds

We characterize conformally flat spaces as the only compact self-dual manifolds which are U(1)-equivariantly and conformally decomposable into two complete self-dual Einstein manifolds with common conformal infinity. A geometric characterization of such conformally flat spaces is also given.     

Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in \mathbbR⁴{\mathbb{R}^4} . The latter condition is different from those in the earlier works of others.     

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in Graham and Lee (Adv Math 87:186–255, 1991), Lee (Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds, 2006), and Biquard (Surveys in Differential Geometry: Essays on Einstein Manifolds, 1999) of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.     

Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature <Emphasis Type="Italic">R</Emphasis> ≥ −<Emphasis Type="Italic">n</Emphasis> (<Emphasis Type="Italic">n</Emphasis> − 1)

In this paper, we use the normalized Ricci–DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ ?n (n ? 1) and also the rigidity result when certain relative volume is zero.     

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S ¹⁵ → S ⁸, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S ¹ subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in ${\mathbb{R}^4}$ . The latter condition is different from those in the earlier works of others.     

Three-manifolds with small L 2-norm of traceless-Ricci curvature pinching

In this paper, we consider a fourth-order gradient flow of the quadratic Riemannian functional ɛ of traceless Ricci curvature on closed 3 -manifolds with a fixed conformal class. We show that the L ²-curvature pinching locally conformally flat 3-manifolds can be deformed to space forms through such gradient flow. More precisely, for the suitable small initial energy functional ɛ, the gradient flow exists for all times and converges smoothly to space forms as the time goes to infinity. As a consequence, we prove the stability for any background metric whose such gradient flow converges to an Einstein metric.Mathematics Subject Classifications (2000): Primary: 53C21; Secondary: 58JOS.Communicated by: Claude LeBrun (Stony Brook)     

Examples of Einstein manifolds in odd dimensions

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kähler–Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano Kähler–Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.