Constant Negative Scalar Curvature Research Articles

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant.

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled [Formula: see text]-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.

Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

Almost Kähler metrics and pp-wave spacetimes

We establish a one-to-one correspondence between a class of strictly almost Kähler metrics on the one hand and Lorentzian pp-wave spacetimes on the other; the latter metrics are well known in general relativity, where they model radiation propagating at the speed of light. Specifically, we construct families of complete almost Kähler metrics by deforming pp-waves via their propagation wave vector. The almost Kähler metrics we obtain exist in all dimensions \(2n \ge 4\), and are defined on both \(\mathbb {R}^{2n}\) and \({\mathbb {S}}^1\times {\mathbb {S}}^1 \times M\), where M is any closed almost Kähler manifold; they are not warped products, they include noncompact examples with constant negative scalar curvature, and all of them have the property that their fundamental 2-forms are also co-closed with respect to the Lorentzian pp-wave metric. Finally, we further deepen this relationship between almost Kähler and Lorentzian geometry by utilizing Penrose’s “plane wave limit,” by which every spacetime has, locally, a pp-wave metric as a limit: using Penrose’s construction, we show that in all dimensions \(2n \ge 4\), every Lorentzian metric admits, locally, an almost Kähler metric of this form as a limit.

The linear stability of the n + 1 dimensional FLRW spacetimes

Here we prove the linear stability of a family of ‘n + 1’-dimensional cosmological models of general relativity locally isometric to the Friedmann Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. We show that the solutions to the linearized Einstein–Euler field equations around a class of FLRW metrics with compact spatial topology (negative Einstein spaces and in particular hyperbolic for n = 3) arising from regular initial data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish the result, we express the Einstein–Euler system in constant mean extrinsic curvature spatial harmonic gauge and linearize about the chosen FLRW background. Utilizing a Hodge decomposition of the fluid’s n-velocity one-form, the linearized system becomes elliptic–hyperbolic (and non-autonomous) in the CMCSH gauge facilitating an application of an energy type argument. Utilizing the estimates derived from the associated elliptic equations, we first prove the uniform boundedness of an energy functional (controlling an appropriate norm of the data) in the expanding direction. Utilizing the uniform boundedness, we later obtain a sharp decay estimate which suggests accelerated expansion (for Λ > 0) of this particular Universe model may be sufficient to control the non-linearities (including possible shock formation) of the Einstein–Euler system in a potential future proof of the fully non-linear stability. In addition, the rotational and harmonic parts of the fluid’s n-velocity field only couple to the remaining degrees of freedom in higher orders, which once again indicates a straightforward extension of current analysis to the fully non-linear setting in the sufficiently small data limit. In addition, our results require a certain integrability condition on the expansion factor and a suitable range of the adiabatic index γ ad ( i.e. in the physically relevant ‘3 + 1’ Universe) if the barotropic equation of state p = (γ ad − 1)ρ is chosen.

Almost Non-negative Scalar Curvature on Riemannian Manifolds Conformal to Tori

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in $C^1$. The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture.

Metrics of Eguchi–Hanson types with the negative constant scalar curvature

We construct two types of Eguchi–Hanson metrics with the negative constant scalar curvature. The type I metrics are Kähler. The type II metrics are ALH whose total energy can be negative.

Metrics of Horowitz–Myers type with the negative constant scalar curvature

We construct a one-parameter family of complete metrics of Horowitz-Myers type with the negative constant scalar curvature. We also verify a positive energy conjecture of Horowitz-Myers for these metrics.

Attractors of the ‘n + 1’ dimensional Einstein-Λ flow

Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of the ‘n + 1’ dimensional vacuum Einstein equations in the presence of a positive cosmological constant Λ. The future stability of vacuum solutions in the small data and zero cosmological constant limit has been studied previously for both ‘3 + 1’ and higher dimensional spacetimes. However, with the advent of dark energy driven accelerated expansion of the Universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy for the vacuum Einstein equations. Such Einsteinian evolution is here designated as the ‘Einstein-Λ’ flow. We study the background solutions of this ‘Einstein-Λ’ flow in ‘n + 1’ dimensional spacetimes in constant mean curvature spatial harmonic gauge, n ⩾ 3 and establish both linear and non-linear stability of such solutions. In the cases of number of spatial dimensions being strictly greater than 3, the finite dimensional Einstein moduli spaces form the centre manifolds of the dynamics. A suitable shadow gauge condition [Andersson L and Moncrief V 2011 J. Differ. Geom. 89 1–47] is implemented in order to treat these cases. In addition, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including Λ(>0). We construct a Lyapunov functional (controlling a suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the data and prove its decay in the direction of cosmological expansion utilizing the structure of the non-autonomous terms and smallness assumption on the data. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the space of constant negative scalar curvature metrics sufficiently close to and containing the Einstein moduli space (a point for n = 3 and a finite dimensional space for n > 3), which has significant consequences for the cosmic topology while restricting to the case of n = 3.

Geometric aspect of three-dimensional Chern–Simons-like gravity

In this paper, we detailed study the geometric aspect of Chern–Simons-like gravity in three dimensions. The vacua are provided by three dimensional conformally flat manifold, which admit a special configuration, a two dimensional system (M2,h,Φ) consisting of metric h and scalar field Φ, by dimensional reduction. For this system we define the quasi-local mass. An interesting observation is that this system contains certain two dimensional dilaton gravity at the classical level. Via AdS/CFT, we check the Weyl anomaly and diffeomorphism anomaly for the boundary theory. We study the linearization of Chern–Simons-like gravity around the background with constant scalar curvature via linear Cotton tensor. If the background manifold is of positive constant scalar curvature, we show that there is no solutions for linearized vacuum equation. To find the solutions of the linear gravity with respect to the background with negative constant scalar curvature, we need to solve some Schrödinger-type equations. Related to supersymmetric quantum mechanics, one can find some exact solutions and some dualities between different components or modes of solutions. And it is also can be related to Seiberg–Witten theory via Picard–Fuchs equation in terms of WKB approximation. We discuss the ADM-type charge under the context of Chern–Simons-like gravity. Finally, we extend the Chern–Simons-like gravity to the supermanifolds by embedding the structure group of three-manifold into the body of othosymplectic supergroup.

The unified soliton system as the AdS2 system

We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with a constant negative scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.

Conformally flat Riemannian manifolds with finite $$L^p$$ L p -norm curvature

Let \((M^n, g)(n\ge 3)\) be an n-dimensional complete, simply connected, locally conformally flat Riemannian manifold with constant scalar curvature S. Denote by T the trace-free Ricci curvature tensor of M. The main result of this paper states that T goes to zero uniformly at infinity if for \(p\ge \frac{n}{2}\), the \(L^{p}\)-norm of T is finite. As applications, we prove that \((M^n, g)\) is compact if the \(L^{p}\)-norm of T is finite and S is positive, and \((M^n, g)\) is scalar flat if \((M^n, g)\) is a noncompact manifold with nonnegative constant scalar curvature and the \(L^{p}\)-norm of T is finite. We prove that \((M^n, g)\) is isometric to a sphere if S is positive and the \(L^{p}\)-norm of T is pinched in [0, C), where C is an explicit positive constant depending only on n, p and S. Finally, we prove an \(L^{p}(p\ge \frac{n}{2})\)-norm of T pinching theorem for complete, simply connected, locally conformally flat Riemannian manifolds with negative constant scalar curvature.

Uniqueness of conformal Ricci flow using energy methods

We analyze an energy functional associated to Conformal Ricci Flow along closed manifolds with constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of both the metric and the pressure function along Conformal Ricci Flow.

On a projectively invariant distance on Finsler spaces of constant negative Ricci scalar

In this work, an intrinsic projectively invariant distance is used to establish a new approach to the study of projective geometry in a Finsler space. It is shown that the projectively invariant distance previously defined is a constant multiple of the Finsler distance when the manifold in question is both forward and backward complete. As a consequence, two projectively related complete Einstein Finsler spaces with constant negative scalar curvature are homothetic. Evidently, this will be true for Finsler spaces of constant flag curvature as well.

Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in three, four and six dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in three-dimensional spaces because of the link to General Relativity. In particular, all three-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz–York method of finding initial data for Einstein equations.

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles S r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\), assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I G and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifolds TM and S r M.

Seiberg--Witten-like equations on $5$-dimensional contact metric manifolds

In this paper, we write Seiberg--Witten-like equations on contact metric manifolds of dimension 5. Since any contact metric manifold has a Spinc-structure, we use the generalized Tanaka--Webster connection on a Spinc spinor bundle of a contact metric manifold to define the Dirac-type operators and write the Dirac equation. The self-duality of 2-forms needed for the curvature equation is defined by using the contact structure. These equations admit a nontrivial solution on 5-dimensional strictly pseudoconvex CR manifolds whose contact distribution has a negative constant scalar curvature.

ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.