Critical Point Equation Research Articles

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.

Critical point equation within the framework of various contact metric manifolds

The aim of the present paper is to study the critical point equation (shortly CPE) conjecture within the framework of various contact metric manifolds. First we establish that Kenmotsu manifold satisfying the CPE either becomes an Einstein manifold or the derivative of potential function along characteristic vector field satisfy a certain relation on the distribution of η. Next we study CPE on (κ, μ)'-almost Kenmotsu manifold and obtain that the manifold is Einstein. Later in case of 3-dimensional trans-Sasakian manifold, we get that either the manifold becomes α-Sasakian or it becomes Einstein. Finally we give examples of 3-dimensional trans-Sasakian manifold and (κ ,μ)'-almost Kenmotsu manifold to verify our outcomes.

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

In Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022), an estimate for skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: Most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature (Xu and Ye in Invent Math 33(2):64, 2022) and in connection with the Besse conjecture (Yun and Hwang in Critical point equation on three-dimensional manifolds and the Besse conjecture). In the present note, we point out an issue in the argument proposed in Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022) and we provide a counterexample to the estimate.

Certain Paracontact Metrics Satisfying the Critical Point Equation

The aim of this paper is to study theCPE (Critical Point Equation) on some paracontact metric manifolds.First, we prove that if a para-Sasakian metric satisfies the CPE,then it is Einstein with constant scalar curvature -2n(2n+1). Next,we prove that if $(\kappa,\mu)$-paracontact metric satisfies theCPE, then it is locally isometric to the product of a flat$(n+1)$-dimensional manifold and $n$-dimensional manifold ofnegative constant curvature$-4$.

A study of critical point equation and *-critical point equation on Sasakian manifolds

In this paper, we initiate the study of critical point equation and *-critical point equation within the substructure of Sasakian manifolds. We prove that if a compact Sasakian manifold admits CPE, then either the manifold is Einstein or the potential function is harmonic in an open subset. Later, we demonstrate that if the manifold satisfies *-CPE, it is [Formula: see text]-Einstein and obtain an expression for the *-Ricci tensor. Finally, we construct an example to illustrate the existence of CPE and *-CPE on the Sasakian manifold.

Rigidity of compact almost Ricci solitons and applications of Pohozaev-Schoen identity

In this paper, we establish a condition for a closed almost Ricci soliton to be an Einstein manifold. As a by product of our approach, we present direct proofs of some interesting results about almost Ricci solitons and the Critical Point Equation (CPE) metrics of the total scalar curvature functional.

On the geometry of Einstein-type manifolds with some structural conditions

In this paper, we investigate the geometry of an Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation. We show various rigidity results of Einstein-type manifolds under assumptions of several curvature conditions.

Besse conjecture with positive isotropic curvature

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function f on the manifold that satisfies the following $$\begin{aligned} (1+f)\mathrm{Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. \end{aligned}$$ It has been conjectured that if (g, f) is a solution of the critical point equation, then g is Einstein and so (M, g) is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.

Modification of scaled equation of state to determine the pressure in the CO2 critical region

The object of the research is carbon dioxide and its pressure distribution depending on the range of temperature and density in the region of the critical point. One of the most problematic areas of methods for finding thermodynamic parameters of a real gas is insufficient accuracy in calculations in the places of occurrence and rapid development of fluctuation phenomena, which are inherent in phase transitions of the first and second terms. For a more detailed and accurate description of the nature of the thermodynamic parameters in the region of the critical point, scaling and crossover equations of state were developed. Such equations, due to the presence of regular and scaling parts, allow describing the thermodynamic parameters of a real gas not only directly near the critical point, but also at some distance from it, maintaining a small error relative to experimental data. The article proposes an equation of state, which contains a scaling part described according to the rules of statistical physics, and a regular part in the form of a classical cubic equation of state. The equation is used to calculate the pressure of carbon dioxide in the region around the critical point from 300 K to 305 K. The article proposes a correlation equation for the scaling correction of the regular part (Redlich-Kwong-Aungier model) of the crossover equation of state, which is related to the scaling part the equation of state is a crossover function. The obtained results for the pressure in the critical region showed good agreement with the baseline data. The error relative to the experimental data is halved compared to the original model of the Redlich-Kwong-Aungier equation. The obtained results ensure the applicability of the proposed method in the temperature range from 300 K to 305 K. Due to the simplicity of the form of the regular equation of state and the small number of empirical coefficients for the large-scale equation of state, the obtained method can be used for practical problems of computational hydrodynamics without spending a lot of computing time.

Generalized TAP Free Energy

AbstractWe consider the mixed p‐spin mean‐field spin glass model with Ising spins and investigate its free energy in the spirit of the TAP approach, named after Thouless, Anderson, and Palmer [67]. More precisely, we define and compute the generalized TAP correction, and establish the corresponding generalized TAP representation for the free energy. In connection with physicists’ replica theory, we introduce the notion of generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameters match the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, our result shows that the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards‐Anderson parameter. © 2022 Wiley Periodicals LLC.

Vanishing conditions on Weyl tensor for Einstein-type manifolds

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.

A note on critical point equations on three-dimensional cosymplectic manifolds

The goal of this note is to investigate some properties of the critical point equations on the $3-$dimensional $f-$ cosymplectic manifolds. We obtain some geometric equations on the $3-$dimensional $f-$ cosymplectic manifolds which admit critical point equations. We give a relation between $f$ and $tilde{f}$ for a CPE metric on the three dimensional $f-$cosymplectic manifold to be Einstein. Also we obtain an eigenvalue of the Laplace operator on the $3-$dimensional $f-$ cosymplectic manifolds with CPE metrics.

Critical point equation on almost f-cosymplectic manifolds

PurposeBesse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f-cosymplectic manifolds.Design/methodology/approachThe paper opted the tensor calculus on manifolds to find the solution of the CPE.FindingsIn this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\tilde{f} is Einstein. Next, the authors find that a three dimensional almost f-cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti‐commuting.Originality/valueThe paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.

Sets that maximize probability and a related variational problem

Let be a random variable of a Riemannian manifold. We assume that the C2‐probability density function of exists. This research addresses two variational questions. The first concerns sets that maximize their probability among those that have a fixed volume. We prove that such a set must have a probability density function that is constant along its boundary; equivalently, such a set must be a density level set. We also obtain the equations related to the maximization property (the stability of the solutions). The other variational problem is the inverse of the first question, namely which sets minimize their volume among those sets which have a predetermined probability? The solution of this problem will define a notion of a quantile set. We show that the solutions of both variational problems coincide (the critical point equation and the stability condition). As theoretical applications, we consider a decision‐making problem and fuzzy sets. As practical applications, we first explore how to locate a powerplant, and subsequently develop a model for a distribution of cheetahs.

Critical point equation and closed conformal vector fields

AbstractIn this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non‐trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.

The critical point equation on Kenmotsu and almost Kenmotsu manifolds

In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfies the CPE.

Topological black hole chemistry in massive gravity with power-Maxwell invariant field

In this paper, asymptotically AdS black hole solutions of massive gravity in the presence of nonlinear electromagnetic field arisen from the power theory of Maxwell invariant are investigated and the associated Euclidean on-shell action is presented. Using the Euclidean on-shell action, the gravitational partition function in the canonical ensemble is computed in arbitrary dimensions and then thermodynamic quantities of topological black holes are obtained. By extending the thermodynamic phase space, i.e., treating negative cosmological constant as thermodynamic pressure, the first law of thermodynamics as well as associated Smarr formula are examined. Next, the equation of state of topological black holes is obtained and it is proven that the critical point equation of these solutions can exhibit black hole phase transitions similar to those of van der Waals, van der Waals like and solid/liquid/gas (related to triple point) phase transitions in usual thermodynamic systems. Especially, the van der Waals phase transition is observed in 4 and higher dimensions, van der Waals type phase transition can be seen in 6 and higher dimensions, and phase transitions associated with triple point, i.e., small/intermediate/large black hole phase transition may happen in 6 and higher dimensions.

Critical Point Equation on Almost Kenmotsu Manifolds

We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein.

The Critical Point Equation on 3-dimensional α-cosymplectic Manifolds

The Critical Point Equation on 3-dimensional α-cosymplectic Manifolds

$$*$$-Critical point equation on a class of almost Kenmotsu manifolds

In the present paper, we characterize $$(k,\mu )'$$-almost Kenmotsu manifolds admitting $$*$$-critical point equation. It is shown that if $$(g, \lambda )$$ is a non-constant solution of the $$*$$-critical point equation of a connected non-compact $$(k,\mu )'$$-almost Kenmotsu manifold, then (1) the manifold M is locally isometric to $$\mathbb {H}^{n+1}(-4)$$$$\times $$$$\mathbb {R}^n$$, (2) the manifold M is $$*$$-Ricci flat and (3) the function $$\lambda $$ is harmonic. Finally an illustrative example is presented.