Constant Scalar Curvature Research Articles

Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity

Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability

In Kähler geometry, Calabi extremal metrics serves as a class of more available special metrics than Kähler metrics with constant scalar curvatures, as a generalization of Kähler Einstein metrics. In recent years, Maxwell–Einstein metrics (or conformally Kähler Einstein–Maxwell metrics) appeared as another alternative choice for Calabi extremal metrics. It turns out that some similar metrics defined by Futaki and Ono have similar roles in the Kähler geometry. In this paper, we prove that for some completions of certain line bundles, there is at least one k-generalized Maxwell–Einstein metric defined by Futaki and Ono conformally related to a metric in any given Kähler class for any integer 3≤k≤13.

In this paper, two concircular invariants of a nearly Kähler manifold are considered. It is proved that a nearly Kähler manifold is concircularly flat if and only if the first concircular invariant is zero. A formula for calculating the second concircular invariant is obtained, and a subclass of nearly Kähler manifolds is distinguished, called the class of concircular-paraKähler manifolds. A concircular-paraKähler manifold of zero scalar curvature is isometric to the complex Euclidean space $\mathbb{C}^n$ equipped with the standard Hermitian metric. The class of concircular-paraKähler manifolds of nonzero constant type coincides with the class of six-dimensional proper nearly Kähler manifolds. It is proved that a concircular-paraKähler nearly Kähler manifold is a Riemannian manifold of constant nonnegative scalar curvature. In this case, its scalar curvature is zero if and only if it is a Kähler manifold. A complete local characterization of concircular-paraKähler nearly Kähler manifolds and concircular-recurrent nearly Kähler manifolds is obtained.

This paper investigates a plane-symmetric cosmological model (PSCM) in the context of modified f(R) gravity theory, incorporating both vacuum and non-vacuum scenarios. A perfect fluid is assumed as the matter source. To obtain the solutions, we consider the premise of constant scalar curvature. By applying the conservation law for Einstein's field equation, Tij;j, and the power-law assumption, we retrieve some well-known solutions. We solved the field equations by making a specific assumption that involved a transformation A2B=U . This study explores the physical and kinematic characteristics of specific cosmological models, along with an examination of the statefinder diagnostic—a key tool for analysing the universe’s evolutionary trajectory. The work provides important insights into the behaviour of anisotropic models within the context of modified f(R) gravity. It highlights the interplay between matter distribution and spacetime geometry, particularly emphasizing how assuming a constant scalar curvature aids in simplifying and solving the corresponding field equations. The resulting solutions enhance our understanding of cosmic evolution governed by modified f(R) gravity.

Abstract We introduce a sequence of conformally invariant scalar curvature quantities, defined along the conformal infinity of a conformally compact (CC) manifold, that measure the failure of a CC metric to have constant negative scalar curvature in the interior, i.e. its failure to solve the singular Yamabe problem. Indeed, these ‘CC boundary curvature scalars’ compute canonical expansion coefficients for singular Yamabe metrics. Residues of their poles yield obstructions to smooth solutions to the singular Yamabe problem and thus, in particular, give an alternate derivation of generalized Willmore invariants. Moreover, in a given dimension, the critical CC boundary scalar characterizes the image of a Dirichlet-to-Neumann map for the singular Yamabe problem. We give explicit formulæ for the first five CC boundary curvature scalars required for a global study of four dimensional singular Yamabe metrics, as well as asymptotically de Sitter spacetimes.

Abstract We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino–Schoen-type hyperbolic gluing, partly extending the recently introduced Mao–Oh–Tao gluing method to the hyperbolic setting.

Abstract This research paper seeks to investigate the characteristics of almost Riemann solitons and almost gradient Riemann solitons within the framework of generalized Robertson Walker (GRW) spacetimes that incorporate imperfect fluids. Our study begins by defining specific properties of the potential vector field linked to these solitons. 
 We examine the potential vector field of an almost Riemann soliton on GRW imperfect fluid spacetimes, establishing that it aligns collinearly with a unit timelike torse-forming vector field. This leads us to express the scalar curvature in relation to the structures of soliton and spacetime. Furthermore, we explore the characteristics of an almost gradient Riemann soliton with a potential function $\psi$ across a range of GRW imperfect fluid spacetimes, deriving a formula for the Laplacian of $\psi$. We also categorize almost Riemann solitons on GRW imperfect fluid spacetimes into three types: shrinking, steady, and expanding, when the potential vector field of the soliton is Killing. We prove that a GRW imperfect fluid spacetime with constant scalar curvature and a Killing vector field admits an almost Riemann soliton. Additionally, we demonstrate that if the potential vector field of the almost Riemann soliton is a $\nu(Ric)$-vector, or if the GRW imperfect fluid spacetime is $\mathcal{W}_{2}$-flat or pseudo-projectively flat, the resulting spacetime is classified as a dark fluid.

Abstract This paper investigates the geometric and physical properties of the pseudo-projective curvature tensor in the context of Sequential Doubly Warped Product Manifolds (SDWPM). We derive comprehensive expressions for this tensor’s components and establish precise necessary and sufficient conditions under which a SDWPM becomes pseudo-projectively flat. The analysis extends to special cases involving Einstein manifolds and constant scalar curvature, where we obtain explicit relationships between warping functions. Our findings significantly extend previous results on warped product manifolds, providing new geometric insights with applications to theoretical physics. The physical applications section explores implications for cosmological models, black hole physics, modified gravity theories, and multidimensional spacetimes. We further establish connections between pseudo-projective geometric properties and observable phenomena in cosmology and astrophysics, offering potential experimental tests for the theoretical framework.

Abstract For a small polarised deformation of a constant scalar curvature Kähler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature Kähler metric. In this setting, we reduce K-polystability to the computation of the classical Futaki invariant on the cscK degeneration. Our result holds on specific families and provides local wall-crossing phenomena for the moduli of cscK manifolds when the polarisation varies.

Abstract For , we construct a smooth metric on the standard ‐dimensional sphere such that there exists a sequence of smooth metrics conformal to where each has scalar curvature and their volumes tend to infinity as approaches infinity.

This paper investigates a class of static spacetimes within higher-dimensional ([Formula: see text]) scalar–torsion theories featuring non-minimal derivative coupling and an active scalar potential. The spacetime structure is conformally related to a product space comprising a two-surface and a [Formula: see text]-dimensional submanifold. By analyzing the equations of motion, we demonstrate that the [Formula: see text]-dimensional submanifold must admit constant triplet structures, one of which corresponds to the torsion scalar. This condition allows the equations of motion to be reduced to a single highly nonlinear ordinary differential equation, referred to as the master equation. Our analysis reveals that the solutions to this model generically exhibit at least one naked singularity at the origin, ruling out the possibility of black hole or wormhole configurations. In the asymptotic region, the spacetimes converge to geometries with constant scalar curvature, which, in general, do not satisfy Einstein’s field equations. To further explore the behavior of the solutions, we employ a perturbative approach to linearize the master equation and construct first-order corrections. Additionally, we establish the local and global existence properties of the master equation and rigorously prove the non-existence of regular global solutions for [Formula: see text]. These findings provide valuable insights into the structure and limitations of the scalar–torsion theory under consideration.

In this paper, we first study *-Ricci-Bourguignon solitons (in short, *-RB solitons) and find their geometric characterizations on Sasakian manifolds. We show that if a Sasakian metric g admits a non-trivial *-RB soliton, then it is -homothetically fixed null η-Einstein (transverse Calabi-Yau), and moreover, the soliton vector field is a non-strict infinitesimal contact transformation that leaves the structure tensor φ invariant and is a Jacobi field along trajectories of the Reeb vector field. Next, we explore contact RB almost solitons on almost contact α-cosymplectic 3-manifolds. We establish that such a space must have constant non-positive scalar curvature. It is also shown that a simply connected homogeneous α-cosymplectic 3-manifold admitting a contact RB almost soliton, under some hypothesis, is an uni-modular semidirect product Lie group G of type . Further, if µ ≠ 0, then G is the Lie group equipped with its flat left-invariant cosymplectic structure. And if µ = 0, then G is the abelian Lie group ℝ3 with its flat left-invariant cosymplectic structure.

We prove that the pluriclosed flow preserves the Vaisman condition on compact complex surfaces if and only if the starting metric has constant scalar curvature.

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak β-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak β-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with β=const and equipped with an η-Ricci soliton structure whose potential vector field satisfies certain conditions are η-Einstein manifolds of constant scalar curvature.

Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1

Abstract We show that every regular domain 𝒟 in Minkowski space R n , 1 \mathbb{R}^{n,1} which is not a wedge admits an entire hypersurface whose domain of dependence is 𝒟 and whose scalar curvature is a prescribed constant (or function, under suitable hypotheses) in ( − ∞ , 0 ) (-\infty,0) . Under rather general assumptions, these hypersurfaces are unique and provide foliations of 𝒟. As an application, we show that every maximal globally hyperbolic Cauchy compact flat spacetime admits a foliation by hypersurfaces of constant scalar curvature, generalizing to any dimension previous results of Barbot–Béguin–Zeghib (for n = 2 n=2 ) and Smith (for n = 3 n=3 ).

In the current paper, we have studied almost Schouten solitons and gradient Schouten solitons on an $N(\kappa)$-Contact metric manifold of dimension $(2n+1)$. Besides, we have shown that the almost Schouten soliton does not exist on $N(\kappa)$-Contact metric manifold for $\kappa<1$. In addidtion, it has been proved that the manifold complying with the gradient Schouten solitons only when it is an almost $N(\kappa)$-Contact metric manifold. Moreover,we have determined that a $3$-dimensional $N(\kappa)$-Contact metric manifold admitting a gradient Schouten soliton is either flat or of constant scalar curvature. Finally, an example has been conducted to verify the outcomes.

In this paper we expand on the work of the first author on ambient obstruction solitons, which are self-similar solutions to the ambient obstruction flow. Our main result is to show that any closed ambient obstruction soliton is ambient obstruction flat and has constant scalar curvature. We show, in fact, that the first part of this result is true for a more general extended soliton equation where we allow an arbitrary conformal factor to be added to the equation. We discuss how this implies that, on a compact manifold, the ambient obstruction flow has no fixed points up to conformal diffeomorphism other than ambient obstruction flat metrics. These results are the consequence of a general integral inequality that can be applied to the solitons to any geometric flow. Additionally, we use these results to obtain a generalization of the Bourguignon–Ezin identity on a closed Riemannian manifold and study the converse problem on a closed extended q-soliton. We also study this extended equation further in the case of homogeneous and product metrics.

In this study, we investigated the geometric and physical implications of the W3 curvature tensor within the framework of f(R,G) gravity. We found the sufficient conditions for W3 flat spacetimes with constant scalar curvature to be de Sitter (R>0) or Anti-de Sitter (R<0) models. The properties of isotropic spacetime in the modified gravity framework were also investigated. Furthermore, we explored spacetimes with a divergence-free W3 curvature tensor. The necessary and sufficient condition for a W3 Ricci recurrent and parallel spacetime to transform into an Einstein spacetime was determined. Finally, we analyzed the role of the W3 curvature tensor in black hole thermodynamics within f(R,G) gravity.