Scalar Curvature Research Articles

Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation

We study the optimal partition problem for the prescribed constant Q-curvature equation induced by the higher-order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed Q-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant Q-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of GJMS operators on Ricci solitons, compute the Q-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with a prescribed number of nodal domains on the Koiso–Cao Ricci soliton.

Monotonicity of the p-Green Functions

Abstract On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang [ 43] in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

A quadric ansatz method for a certain class of second order PDEs

We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional Kähler-Einstein equation with symmetry and the one-sided type-D Einstein equation with nonzero scalar curvature. The procedure, which reduces the PDEs to ordinary differential equations (ODEs), involves imposing additional constraints to the form of the ansatz, depending on the exact form of the PDEs. Thus its applicability varies within the class. We identify applicable subclasses of a class of modified Toda equations, one of which includes the Kähler-Einstein equation with symmetry, and give the corresponding reduced ODEs. In addition, we obtain a quadric ansatz reduction for a family of Einstein-Weyl equations in arbitrary dimensions. Lastly, we suggest a particular form of equations for which our procedure appears to be most effective as the aforementioned constraints vanish, and also present some reduction results using another type of ansatz, namely the hyperplane ansatz.

Conformal vector fields on Lie groups: The trans-Lorentzian signature

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p,q). In this paper, we study pseudo-Riemannian Lie groups (G,〈⋅,⋅〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g, then dim⁡h≤max⁡{0,min⁡{p,q}−2}. Secondly, we classify trans-Lorentzian Lie groups (i.e., min⁡{p,q}=2) with non-Killing left-invariant conformal vector fields, and prove that [g,g] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat.

On the moduli space curvature at infinity

We analyse the scalar curvature of the vector multiplet moduli space MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} of type IIA string theory compactified on a Calabi-Yau manifold X. While the volume of MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} is known to be finite, cases have been found where the scalar curvature diverges positively along trajectories of infinite distance. We classify the asymptotic behaviour of the scalar curvature for all large volume limits within MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document}, for any choice of X, and provide the source of the divergence both in geometric and physical terms. Geometrically, there are effective divisors whose volumes do not vary along the limit. Physically, the EFT subsector associated to such divisors is decoupled from gravity along the limit, and defines a rigid N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 field theory with a non-vanishing moduli space curvature Rrigid. We propose that the relation between scalar curvature divergences and field theories decoupled from gravity is a common trait of moduli spaces compatible with quantum gravity.

A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

We study notions of isotopy and concordance for Riemannian metrics on manifolds with boundary and, in particular, we introduce two variants of the concept of minimal concordance, the weaker one naturally arising when considering certain spaces of metrics defined by a suitable spectral ''stability'' condition. We develop some basic tools and obtain a rather complete picture in the case of surfaces.

Einstein–Cartan pseudoscalaron inflation

We study a class of early universe cosmological models based on Einstein–Cartan gravity and including a higher derivative term corresponding to a power of the Holst scalar curvature. The resulting effective action is basically given by General Relativity and an additional neutral pseudoscalar field (the pseudoscalaron), unequivocally related to the corresponding components of the torsion, that necessarily acquire a dynamics. The induced pseudoscalaron potential provides a realistic inflationary phase together with a very rich postinflationary epoch, resulting from the coupling of the pseudoscalaron to ordinary matter.

Families of Metrics with Positive Scalar Curvature on Spectral Sequence Cobordisms

Families of Metrics with Positive Scalar Curvature on Spectral Sequence Cobordisms

Black hole shadows in Einstein-Bel-Robinson gravity* *Supported by the Internal Project (2023/2211) of Excellent Research of the Faculty of Science of Hradec Králové University

Gravity models given by higher-order scalar curvature corrections are believed to bear important consequences. Einstein-Bel-Robinson (EBR) gravity with quartic curvature modification motivated Sajadi et al. to explore static spherically symmetric black hole solutions using perturbative methods. In this study, inspired by their work, we investigate AdS black hole shadows in EBR gravity and demonstrate how the gravity parameter alters the energy emission rate. Finally, we address the same problem in the presence of plasma, because the black holes are thought to be surrounded by a medium that changes the geodesic of photons.

Characterizations of spacetimes admitting critical point equation and f(r)-gravity

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in [Formula: see text]-gravity. For the model [Formula: see text] ([Formula: see text] = constant and [Formula: see text] is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions.

Kropina Metrics with Isotropic Scalar Curvature via Navigation Data

Through an interesting physical perspective and a certain contraction of the Ricci curvature tensor in Finsler geometry, Akbar-Zadeh introduced the concept of scalar curvature for the Finsler metric. In this paper, we show that the Kropina metric is of isotropic scalar curvature if and only if F is an Einstein metric according to the navigation data. Moreover, we obtain the three-dimensional rigidity theorem for an Einstein–Kropina metric.

Sustaining Quasi De-Sitter Inflation with Bulk Viscosity

The de-Sitter spacetime is a maximally symmetric Lorentzian manifold with constant positive scalar curvature that plays a fundamental role in modern cosmology. Here, we investigate bulk-viscosity-assisted quasi de-Sitter inflation, that is the period of accelerated expansion in the early universe during which −H˙≪H2, with H(t) being the Hubble expansion rate. We do so in the framework of a causal theory of relativistic hydrodynamics, which takes into account non-equilibrium effects associated with bulk viscosity, which may have been present as the early universe underwent an accelerated expansion. In this framework, the existence of a quasi de-Sitter universe emerges as a natural consequence of the presence of bulk viscosity, without requiring introducing additional scalar fields. As a result, the equation of state, determined by numerically solving the generalized momentum-conservation equation involving bulk viscosity pressure turns out to be time dependent. The transition timescale characterising its departure from an exact de-Sitter phase is intricately related to the magnitude of the bulk viscosity. We examine the properties of the new equation of state, as well as the transition timescale in the presence of bulk viscosity pressure. In addition, we construct a fluid description of inflation and demonstrate that, in the context of the causal formalism, it is equivalent to the scalar field theory of inflation. Our analysis also shows that the slow-roll conditions are realised in the bulk-viscosity-supported model of inflation. Finally, we examine the viability of our model by computing the inflationary observables, namely the spectral index and the tensor-to-scalar ratio of the curvature perturbations, and compare them with a number of different observations, finding good agreement in most cases.

Some exact solutions of Friedmann cosmological equation

In this paper we present a number of examples of exact solutions for the Friedmann cosmological equation for metric F(R) gravity model. Emphasis was placed on the possibility of obtaining exact time dependences of the main cosmological physical quantities: scale factor, scalar curvature, Hubble rate and function F(R). For this purpose an ansatz was used to reduce the Friedmann equation to an ordinary differential equation for function F=F(H2). This made it possible to obtain a number of exact solutions, both already known and new.

Consequences of Rényi entropy on the thermal geometries and Hawking evaporation of topological dyonic dilaton black hole

Abstract The thermodynamics of black holes (BHs) has had a profound impact on theoretical physics, providing insight into the nature of gravity, the quantum structure of spacetime and the fundamental laws governing the Universe. In this study, we investigate thermal geometries and Hawking evaporation of the recently proposed topological dyonic dilaton BH in anti-de Sitter (AdS) space. We consider Rényi entropy and obtain the relations for pressure, heat capacity and Gibbs free energy and observe that the Rényi parameter and dilaton field play a vital role in the phase transition and stability of the BH. Moreover, we use Weinhold, Ruppeiner and Hendi Panahiyah Eslam Momennia models to evaluate the scalar curvature of the BH and find out that the divergence points of the scalar curvature coincides with the zero of specific heat. Finally, using Stefan–Boltzmann law, we determine that the BH without a dilaton field evaporates far more quickly compared to the dilaton BH in AdS space.

A Green’s Function Proof of the Positive Mass Theorem

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green’s function of an asymptotically flat 3-manifolds. In the same context and for 1<p<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<3$$\\end{document}, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.

On minimality and scalar curvature of CMC triharmonic hypersurfaces in pseudo-Riemannian space forms

On minimality and scalar curvature of CMC triharmonic hypersurfaces in pseudo-Riemannian space forms

Chern Flat and Chern Ricci-Flat Twisted Product Hermitian Manifolds

Let (M1,g) and (M2,h) be two Hermitian manifolds. The twisted product Hermitian manifold (M1×M2f,G) is the product manifold M1×M2 endowed with the Hermitian metric G=g+f2h, where f is a positive smooth function on M1×M2. In this paper, the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold are derived. The necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature are obtained. Under the condition that the logarithm of the twisted function is pluriharmonic, it is proved that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat, if and only if M1,g and M2,h are Chern flat or Chern Ricci-flat, respectively.

The particle dynamics around rotating charged black holes with Weyl corrections

In this article, particle dynamics around charged rotating black holes with Weyl corrections is discussed. First, the metric is studied for its horizon structure and scalar curvature invariants. The effects of Weyl corrections on circular motion of the particles is analyzed around central object. We have shown dependency of effective potential, energy, angular momentum and inner most stable circular orbits on black hole parameters. We have also determined center of mass energy expression for collision of two neutral particles. In the Penrose process of energy extraction, it has been found that efficiency is not that much higher as compared to the other modified rotating black holes.

Parameter space geometry of the quartic oscillator and the double-well potential: classical and quantum description

We compute the quantum metric tensor and its scalar curvature for the anharmonic oscillator for positive and negative quadratic potentials, where the potential displays a double well, employing exact numerical and perturbative procedures. We also introduce a formulation of the classical analog of the quantum metric tensor by using a novel approach based on Fourier series, which is shown to reproduce the relevant quantum features involved in the parameter space. It is remarkable that both the exact quantum treatment and classical formalism recognize the negative oscillator parameter at which the ground state starts to be delocalized in two wells.

B. Y. Chen-Ricci inequalities for anti-invariant Riemannian submersions in Kenmotsu space forms

The aim of the present paper is to analyze sharp type inequalities including the scalar and Ricci curvatures of anti-invariant Riemannian submersions in Kenmotsu space forms Ks(ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{s}(\\varepsilon )$$\\end{document}. We give non-trivial examples for anti-invariant Riemannian submersions, investigate some curvature relations between the total space and fibres according to vertical and horizontal cases of ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. Moreover, we acquire Chen-Ricci inequalities on the kerϑ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ker \\vartheta _{*}$$\\end{document} and (kerϑ∗)⊥\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\ker \\vartheta _{*})^{\\bot }$$\\end{document} distributions for anti-invariant Riemannian submersions from Kenmotsu space forms according to vertical and horizontal cases of ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}.