Ricci-flat Research Articles

Exotic G_2-manifolds

We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricci-flat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3-folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells–Kuiper invariant introduced by the authors in a previous paper.

Decomposable (6, 5)-solutions in 11-dimensional supergravity

This paper presents a series of constructions providing 11-dimensional bosonic supergravity backgrounds. In particular, we treat Lorentzian manifolds given in terms of twisted products of six-dimensional Lorentzian manifolds and five-dimensional Riemannian manifolds. By considering a representative flux 4-form adapted to this setting, we analyse the system of bosonic supergravity equations and describe the corresponding geometric constraints. The new supergravity backgrounds appear for special cases associated to the adapted flux 4-form. For example, we provide a relation of 11-dimensional supergravity with Ricci-isotropic Walker manifolds, and illustrate several results in their terms and in terms of Ricci-flat Riemannian manifolds.

Ricci-Flat Metrics on Vector Bundles Over Flag Manifolds

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group SU(n), for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas–de la Ossa on the conifold, Pando Zayas–Tseytlin on the canonical bundle over mathbb {CP}^1times mathbb {CP}^1, as well as the metrics on canonical bundles over flag manifolds, recently constructed by van Coevering.

Numerical metrics, curvature expansions and Calabi-Yau manifolds

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau manifold. We show that numerical techniques are rather efficient at deciding this issue.

Stability of ALE Ricci-Flat Manifolds Under Ricci Flow

We prove that if an ALE Ricci-flat manifold (M, g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian’s approach in the closed case, we show that integrability holds for ALE Calabi–Yau manifolds which implies that they are dynamically stable.

Classification of left invariant metrics on 4-dimensional solvable Lie groups

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Unitarization from geometry

We study the perturbative unitarity of scattering amplitudes in general dimensional reductions of Yang-Mills theories and general relativity on closed internal manifolds. For the tree amplitudes of the dimensionally reduced theory to have the expected high-energy behavior of the higher-dimensional theory, the masses and cubic couplings of the Kaluza-Klein states must satisfy certain sum rules that ensure there are nontrivial cancellations between Feynman diagrams. These sum rules give constraints on the spectra and triple overlap integrals of eigenfunctions of Laplacian operators on the internal manifold and can be proven directly using Hodge and eigenfunction decompositions. One consequence of these constraints is that there is an upper bound on the ratio of consecutive eigenvalues of the scalar Laplacian on closed Ricci-flat manifolds with special holonomy. This gives a sharp bound on the allowed gaps between Kaluza-Klein excitations of the graviton that also applies to Calabi-Yau compactifications of string theory.

The ciconia metric on the tangent bundle of an almost Hermitian manifold

We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of K\"ahlerian Ricci-flat manifolds in four real dimensions.

On Expansions of Ricci Flat ALE Metrics in Harmonic Coordinates About the Infinity

In this paper, we study the expansions of Ricci flat metrics in harmonic coordinates about the infinity of ALE (asymptotically local Euclidean) manifolds.

New method to generate exact scalar-tensor solutions

A new method is proposed, that establishes a one to one correspondance between the whole set of static axially symmetric vacuum GR solutions and a specific class of stationary axially symmetric scalar-Einstein solutions having a given mass and a given angular momentum. The method explicitly takes advantage of the Kerr metric Ricci flatness. This also results in a class of stationary axially symmetric vacuum, ie Kerrlike, Brans-Dicke solutions. A particular solution, that is asymptotically flat, is more closely considered. It converges to Kerr for a vanishing scalar charge, but fails to converge to the Fisher-Janis-Newman-Winicour solution for a vanishing "rotation parameter". This solution exhibits a naked singularity having a ringlike structure.

Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on $G/K$.

Generalized Ricci flow II: Existence for complete noncompact manifolds

In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points are Ricci-flat metrics. Modifying the argument used by Shi and List, we prove the short time existence and higher order derivatives estimates.

On the Local Extension of Killing Vector Fields in Electrovacuum Spacetimes

We revisit the problem of extension of a Killing vector field in a spacetime which is solution to the Einstein-Maxwell equation. This extension has been proved to be unique in the case of a Killing vector field which is normal to a bifurcate horizon by Yu. Here we generalize the extension of the vector field to a strong null convex domain in an electrovacuum spacetime, inspired by the same technique used by Ionescu-Klainerman in the setting of Ricci flat manifolds. We also prove a result concerning non-extendibility: we show that one can find local, stationary electrovacuum extension of a Kerr-Newman solution in a full neighborhood of a point of the horizon (that is not on the bifurcation sphere) which admits no extension of the Hawking vector field. This generalizes the construction by Ionescu-Klainerman to the electrovacuum case.

A new incomplete Ricci-flat metric

Classical metrics (Eguchi–Hanson, Taub–NUT, Fubini–Study) satisfy a certain system of ODE that we introduced. By studying this system we found new Ricci-flat incomplete Riemannian metric in explicit formula and described its behavior at infinity and in the singular point.

Stability of Ricci de Turck flow on singular spaces

In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Levi-Civita Ricci-flat metrics on compact complex manifolds

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces or Hopf surfaces.

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or Ln2-norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem.

On stationary solutions to the vacuum Einstein field equations

We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation $Ric(g_{M})=\lambda g_{M}$ with nonnegative cosmological constant $\lambda\geq 0$ is flat. When dim $\geq 5$, if the spacetime is assumed to be static additionally, we prove that its universal cover splits isometrically as a product of a Ricci flat Riemannian manifold and a real line.

Projective Ricci flat spherically symmetric Finsler metrics

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we characterize projective Ricci flat spherically symmetric Finsler metrics. Under a certain condition, we prove that a projective Ricci flat spherically symmetric Finsler metric must be a Douglas metric. Moreover, we construct a class of new nontrivial examples on projective Ricci flat Finsler metrics.

Phase Transition of Kähler–Einstein Metrics via Moment Maps

We study the phase transition of Kahler Ricci-flat metrics on some open Calabi–Yau spaces with the help of the images of moment maps of natural torus actions on these spaces.