The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four dimensional smooth manifold $M$ in presence of field $φ$, that in this context is given by a smooth map with source $M$ and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called $φ$-Bach tensor, that in absence of the field $φ$ reduces to the classic Bach tensor. We provide a variational characterization for $φ$-Bach flat manifolds and we do the same also for harmonic-Einstein manifolds, i.e., solutions of the Einstein field equations with source the conservative field $φ$. We take the opportunity to discuss a generalization of some related topics: the Yamabe problem, the image of the scalar curvature map, warped product solutions and static manifolds.
The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold $(M^{n},\,g)$ with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstein fiber. In addition, the fiber has constant curvature if $n = 4.$
In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension $n \geq 4$ is a finite quotient of a warped product with $(n-1)$-dimensional Einstein fiber. The fiber has constant curvature if $n=4$.
In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations in general relativity. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kaehler metrics are solutions to these equations, although, contrary to the Einstein-Maxwel…
It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant $Q$-curvat…
On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will also be Einstein. It was shown that this conjecture is true when $M$ together with a critical metric has harmonic curvature or the metric is Bach flat. In this paper, we tried to prove this conjecture with a divergence-free Bach tensor.
On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture, proposed in 1987 by Besse, has not been resolved except when $M$ has harmonic curvature or the metric is Bach flat. In this paper, we prove some gap properties under divergence-free Bach tensor condition for $n\geq 5$, and a simil…
We prove that an n($\geq$ 4)-dimensional compact Bach-flat manifold with positive constant $σ_2$ is an Einstein manifold, provided that its Weyl curvature satisfies a suitable pinching condition.
This paper presents conformal invariants for Riemannian manifolds of dimension greater than or equal to four whose vanishing is necessary for a Riemannian manifold to be conformally related to an Einstein space. One of the invariants is a modification of the Cotton tensor, the other is a $n$--dimensional version of the Bach tensor. In general both tensors are smooth only on an open and dense subset of $M$, but this subset is invariant under conformal transformations. Moreover, we generalize the main result of "Conformal Einstein Spaces in $N$--Dimensions" published in \emph{Ann. Global Anal. G…
We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results are known for Einstein metrics, but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.
On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat.
We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm. Compared with the complete non-compact case done by Kim, we apply a different method to achieve these results. 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.
In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.
First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and quasi-Einstein ($η$-Einstein) conditions. Finally, we provide local classification and characterization of a semi-Riemannian (including the 4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic conformal (including closed) vector field, in terms of Petrov types and Bach tensor.
We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of the full divergence of the Cotton tensor for static metrics and metrics satisfying the critical point equation.
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to eith…
We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.
A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know \emph{a priori} the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain: (i) a new proof of the stability of Einstein-Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global sta…
We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.
We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our construction is reduced to an ordinary differential equation called Duffing equation. An ODE for Bach-flat metrics on Hirzebruch surfaces with large isometry group is also derived.
In this note we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the TT-tensor is small in $L^2$.
The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods.
We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations.
We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product construction.
We establish a regularity theorem for the Harmonic - Einstein Equation. As a byproduct of the local regularity, we also have a compactness theorem on Harmonic - Einstein equation. The method is mainly the Moser iteration technique which has been used and developed by \cite{BKN89}, \cite{Tian90}, \cite{TV05a} and others.
We introduce some generalizations of the Hermitian-Einstein equation for diagonal harmonic metrics on cyclic Higgs bundles, including a generalization using subharmonic functions. When the coefficients are all smooth, we prove the existence, uniqueness, and convergence of the solution of their heat equations with Dirichlet boundary conditions. We also generalize two inequality estimates for solutions of the Hermitian-Einstein equation for cyclic Higgs bundles.
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature $S \not= 0$ that carry a non-trivial solution of the Einstein-Dirac equation.
We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold $M^n$ admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics $\barη$ on $M^n \times {\mathbb R}$ such that $(M^n \times {\mathbb R}, \barη)$ admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints.
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and only if the manifold is strongly projectively flat.
In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field syst…
As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed.
We considered an extension of the standard functional for the Einstein-Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler-Lagrange equations provide a new type of Einstein-Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein-Dirac system called the {\it CL-Einstein-Dirac equation of type II} (see Definition 3.1).
We apply the technique of integrable extensions to the symmetry pseudo-group of the dKP-hyper CR interpolating equation. This allows us to find a covering for this equation and to construct multi-valued Einstein-Weyl structures.
