Showing papers on "Scalar curvature published in 2023"

Four Lectures on Scalar Curvature

Abstract: We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures

Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

Abstract: We show that if $N$ is a closed manifold of dimension $n=4$ (resp. $n=5$) with $\pi_2(N) = 0$ (resp. $\pi_2(N)=\pi_3(N)=0$) that admits a metric of positive scalar curvature, then a finite cover $\hat N$ of $N$ is homotopy equivalent to $S^n$ or connected sums of $S^{n-1}\times S^1$. Our approach combines recent advances in the study of positive scalar curvature with a novel argument of Alpert--Balitskiy--Guth. Additionally, we prove a more general mapping version of this result. In particular, this implies that if $N$ is a closed manifold of dimensions $4$ or $5$, and $N$ admits a map of nonzero degree to a closed aspherical manifold, then $N$ does not admit any Riemannian metric with positive scalar curvature.

Conjectures on Convergence and Scalar Curvature

Abstract: Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}. We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above. In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.

On the space of riemannian metrics satisfying surgery stable curvature conditions

Abstract: We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.

Spectral Torical Band Inequalities and Generalizations of the Schoen–Yau Black Hole Existence Theorem

Abstract: Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, $\mu $-bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary, we generalize the classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size.

Manifolds with nonnegative curvature operator of the second kind

Abstract: We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa's conjecture under a weaker assumption.

First Natural Connection on Riemannian Π-Manifolds

Abstract: A natural connection with torsion is defined, and it is called the first natural connection on the Riemannian Π-manifolds. Relations between the introduced connection and the Levi--Civita connection are obtained. Additionally, relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar curvatures in the main classes of a classification of Riemannian Π-manifolds are presented. An explicit example of dimension five is provided.

Volume comparison with respect to scalar curvature

Abstract: In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we give a partial answer to a conjecture of Bray and recover a result of Besson, Courtois and Gallot, which partially confirms a conjecture of Schoen about closed hyperbolic manifold. Applying analogous techniques, we obtain a different proof of a local rigidity result due to Dai, Wang and Wei, which shows it admits no metric with positive scalar curvature near strictly stable Ricci-flat metrics.

Boundary Conditions for Scalar Curvature

Abstract: Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

Cosmology from Non‐Minimal Geometry‐Matter Coupling

Abstract: We construct a cosmological model from the inception of the Friedmann-Lemâitre-Robertson-Walker metric into the field equations of the f ( R , L m ) $f(R,L_m)$ gravity theory, with R being the Ricci scalar and L m $L_m$ being the matter lagrangian density. The formalism is developed for a particular f ( R , L m ) $f(R,L_m)$ function, namely R / 16 π + ( 1 + σ R ) L m $R/16\pi +(1+\sigma R)L_{m}$ , with σ being a constant that carries the geometry-matter coupling. Our solutions are remarkably capable of evading the Big-Bang singularity as well as predict the cosmic acceleration with no need for the cosmological constant, but simply as a consequence of the geometry-matter coupling terms in the Friedmann-like equations.

Doubling the equatorial for the prescribed scalar curvatureproblem on ${\mathbb{S}}^N$

Abstract: Abstract Note: Please see pdf for full abstract with equations. We consider the prescribed scalar curvature problem on S N Δ SN v −N(N − 2)/2 v + ​​ K̃ (y)v N+2/N−2 = 0 on S N , v > 0 in S N , under the assumptions that the scalar curvature ​ K̃ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov-Schmidt reduction method.

Study of compact stars in $${\mathcal {R}}+ \alpha {\mathcal {A}}$$ gravity

Abstract: Abstract The main goal of this work is to provide a comprehensive study of relativistic structures in the context of recently proposed $${\mathcal {R}}+ \alpha {\mathcal {A}}$$ R + α A gravity, where $${\mathcal {R}}$$ R is the Ricci scalar, and $${\mathcal {A}}$$ A is the anti-curvature scalar. For this purpose, we examine a new classification of embedded class-I solutions of compact stars. To accomplish this goal, we consider an anisotropic matter distribution for $${\mathcal {R}}+ \alpha {\mathcal {A}}$$ R + α A gravity model with static spherically symmetric spacetime distribution. Due to highly non-linear nature of field equations, we use the Karmarkar condition to link the $$g_{rr}$$ g rr and $$g_{tt}$$ g tt components of the metric. Further, we compute the values of constant parameters using the observational data of different compact stars. It is worthy to mention here that we choose a set of twelve important compact stars from the recent literature namely $$4U~1538{-}52$$ 4 U 1538 - 52 , $$SAX~J1808.4{-}3658$$ S A X J 1808.4 - 3658 , $$Her~X{-}1$$ H e r X - 1 , $$LMC~X{-}4$$ L M C X - 4 , $$SMC~X{-}4$$ S M C X - 4 , $$4U~1820{-}30$$ 4 U 1820 - 30 , $$Cen~X{-}3$$ C e n X - 3 , $$4U~1608{-}52$$ 4 U 1608 - 52 , $$PSR~J1903{+}327$$ P S R J 1903 + 327 , $$PSR~J1614{-}2230$$ P S R J 1614 - 2230 , $$Vela~X{-}1$$ V e l a X - 1 , $$EXO~1785{-}248$$ E X O 1785 - 248 . To evaluate the feasibility of $${\mathcal {R}}+ \alpha {\mathcal {A}}$$ R + α A gravity model, we conduct several physical checks, such as evolution of energy density and pressure components, stability and equilibrium conditions, energy bounds, behavior of mass function and adiabatic index. It is concluded that $${\mathcal {R}}+ \alpha {\mathcal {A}}$$ R + α A gravity supports the existence of compact objects which follow observable patterns.

Geometry of almost contact metrics as an almost *-η-Ricci-Bourguignon solitons

Abstract: In this paper, we give some characterizations by considering almost ∗-[Formula: see text]-Ricci–Bourguignon soliton as a Kenmotsu metric. It is shown that if a Kenmotsu metric endows a ∗-[Formula: see text]-Ricci–Bourguignon soliton, then the curvature tensor [Formula: see text] with the soliton vector field [Formula: see text] is given by the expression [Formula: see text] Next, we show that if an almost Kenmotsu manifold such that [Formula: see text] belongs to [Formula: see text]-nullity distribution where [Formula: see text] acknowledges a ∗-[Formula: see text]-Ricci–Bourguignon soliton satisfying [Formula: see text], then the manifold is Ricci-flat and is locally isometric to [Formula: see text]. Moreover if the metric admits a gradient almost ∗-[Formula: see text]-Ricci–Bourguignon soliton and [Formula: see text] leaves the scalar curvature [Formula: see text] invariant on a Kenmotsu manifold, then the manifold is an [Formula: see text]-Einstein. Also, if a Kenmotsu metric represents an almost ∗-[Formula: see text]-Ricci–Bourguignon soliton with potential vector field [Formula: see text] is pointwise collinear with [Formula: see text], then the manifold is an [Formula: see text]-Einstein.

Doubling of Asymptotically Flat Half-spaces and the Riemannian Penrose Inequality

Abstract: Building on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $\Sigma\subset M$ and mass $m\in\mathbb{R}$. If $3\leq \dim(M)\leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ m\geq\left(\frac{1}{2}\right)^{\frac{n}{n-1}}\,\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $\partial M$ is not totally geodesic, we show how to construct local perturbations of $(M,g)$ that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of $(M,g)$ is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where $\dim(M)=3$ and $\Sigma$ is a connected free boundary hypersurface.

Global monopoles in the extended Gauss-Bonnet gravity

Abstract: We discuss self-gravitating global O(3) monopole solutions associated with the spontaneous breaking of O(3) down to a global O(2) in an extended Gauss-Bonnet theory of gravity in ($3+1$) dimensions, in the presence of a nontrivial scalar field $\mathrm{\ensuremath{\Phi}}$ that couples to the Gauss-Bonnet higher curvature combination with a coupling parameter $\ensuremath{\alpha}$. We obtain a range of values for $\ensuremath{\alpha}<0$ (in our notation and conventions), which are such that a global (Israel type) matching is possible of the space-time exterior to the monopole core $\ensuremath{\delta}$ with a de Sitter interior, guaranteeing the positivity of the Arnowitt-Deser-Misner (ADM) mass of the monopole, which, together with a positive core radius $\ensuremath{\delta}>0$, are both dynamically determined as a result of this matching. It should be stressed that in the general relativity (GR) limit, where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, and $\mathrm{\ensuremath{\Phi}}\ensuremath{\rightarrow}\text{constant}$, such a matching yields a negative ADM monopole mass, which might be related to the stability issues the [Barriola-Vilenkin (BV)] global monopole of GR faces. Thus, our global monopole solution, which shares many features with the BV monopole, such as an asymptotic-space-time deficit angle, of potential phenomenological/cosmological interest, but has, par contrast, a positive ADM mass, has a chance of being a stable configuration, although a detailed stability analysis is pending.

Beyond Schwarzschild–de Sitter spacetimes. III. A perturbative vacuum with nonconstant scalar curvature in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> gravity

Abstract: In violation of the generalized Lichnerowicz theorem advocated by Nelson and others, quadratic gravity admits vacua with non-constant scalar curvature. In a recent publication [Phys. Rev. D 106, 104004 (2022)], we revitalized a program that Buchdahl originated but prematurely abandoned circa 1962 [Nuovo Cimento 23, 141 (1962)], and uncovered a novel exhaustive class of static spherically symmetric vacua for pure $R^2$ gravity. The Buchdahl-inspired metrics we obtained therein are exact solutions which exhibit non-constant scalar curvature. A product of fourth-order gravity, the metrics entail a new (Buchdahl) parameter $k$ which allows the Ricci scalar to vary on the manifold. The metrics are able to defeat the generalized Lichnerowicz theorem by evading an overly strong restriction on the asymptotic falloff in the Ricci scalar as assumed in the theorem. The Buchdahl parameter $k$ is a new characteristic of pure $R^2$ gravity, a higher-derivative theory. By venturing that the Buchdahl parameter should be a universal hallmark of higher-derivative gravity at large, in this paper we seek to extend the concept to the quadratic action $R^2+\gamma\,(R-2\Lambda)$. We are able to determine that the quadratic field equation admits a perturbative vacuo that is valid up to the order $O(k^2)$. Conforming with our guiding intuition, the Ricci scalar is non-constant, including the asymptotically flat case, as long as $k eq0$ and $\gamma eq0$. The existence of such an asymptotically flat vacuo with non-constant scalar curvature defeats the generalized Lichnerowicz theorem in its entirety. Our finding thus warrants restoring the $R^2$ term in the full quadratic action, $\gamma\,R+\beta\,R^2-\alpha\,C^{\mu u\rho\sigma}C_{\mu u\rho\sigma}$, when applying the L\"u-Perkins-Pope-Stelle ansatz. Implications to the L\"u-Perkins-Pope-Stelle solution are discussed herein.

Beyond Schwarzschild–de Sitter spacetimes. II. An exact non-Schwarzschild metric in pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> gravity and new anomalous properties of <mml:math xmlns:mml="http

Abstract: In our recent publication (Phys. Rev. D 106, 104004 (2022)), we advanced a program that Buchdahl originated but prematurely abandoned circa 1962 (Nuovo Cimento 23, 141 (1962)). Therein we obtained an exhaustive class of metrics that constitute the branch of non-trivial solutions to the pure $R^{2}$ field equation in vacuo. The Buchdahl-inspired metrics in general possess non-constant scalar curvature, thereby defeating the generalized Lichnerowicz theorem previously advocated for quadratic gravity. We found that the said theorem makes an overly strong assumption on the asymptotic falloff in the spatial derivatives of the Ricci scalar, rendering it violable against the Buchdahl-inspired metrics. In this paper, we shall further extend our work mentioned above by showing that, within the class of Buchdahl-inspired metrics, the asymptotically flat member takes on an exact closed analytical expression. The new metric is characterized by a horizon radius $r_{\text{s}}$ and the Buchdahl parameter $k$, the latter of which arises via the higher-derivative nature of $R^{2}$ gravity. For $k=0$, the new metric recovers the classic Schwarzschild metric. Equipped with the exact expression of the new metric, we shall analytically construct the Kruskal-Szekeres diagram for pure $R^{2}$ spacetime. We find that the Buchdahl parameter $k$ fundamentally modifies the properties of $R^{2}$ spacetime structures in a variety of ways.

Eigenvalue Estimates on Quaternion-Kähler Manifolds

Abstract: We prove lower bound estimates for the first nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold. For the closed or Neumann case, the lower bounds depend on dimension, diameter, and lower bound of scalar curvature, and they are derived as the large time implication of the modulus of continuity estimates for solutions of the heat equation. For the Dirichlet case, we establish lower bounds that depend on dimension, inradius, lower bound of scalar curvature, and lower bound of the second fundamental form of the boundary, via a Laplace comparison theorem for the distance to the boundary function.

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> corrections to the black string instability and the boosted black string

Abstract: We consider a perturbative Gauss-Bonnet term supplementing the Einstein-Hilbert action and evaluate its effect on the spectrum of the scalar mode that triggers the Gregory-Laflamme instability of black strings in five-dimensional general relativity. After studying some properties of the static black string, we provide the correction to the Lichnerowicz operator up to $\mathcal{O}({\ensuremath{\alpha}}^{2})$. For the scalar mode of the gravitational perturbation, we find a master variable and study its spectrum, providing an analysis of the regime of validity of our scheme. We show that the instability persists under the inclusion of the ${R}^{2}$ correction and that the critical wavelength increases with the value of $\ensuremath{\alpha}/{r}_{+}^{2}\ensuremath{\ll}1$. We also construct the boosted black strings and compute the correction to the mass, the momentum, and the tension due to the higher curvature term. The presence of the dimensionful coupling $\ensuremath{\alpha}$ spoils the validity of the Smarr relation, which is the gravitational version of the Euler relation that must hold for every homogeneous, thermodynamic system. We show that this identity can be restored by working in an extended thermodynamic setup that includes variations of the Gauss-Bonnet coupling.

Quasi-topological gravities on general spherically symmetric metric

Abstract: A bstract In this work we study a more restricted class of quasi-topological gravity theories where the higher curvature terms have no contribution to the equation of motion on general static spherically symmetric metric where g tt g rr ≠ constant. We construct such theories up to quintic order in Riemann tensor and observe an important property of these theories: the higher order term in the Lagrangian vanishes identically when evaluated on the most general non-stationary spherically symmetric metric ansatz. This not only signals the higher terms could only have non-trivial effects when considering perturbations, but also makes the theories quasi-topological on a much wider range of metrics. As an example of the holographic effects of such theories, we consider a general Einstein-scalar theory and calculate it’s holographic shear viscosity.