Scalar Curvature Research Articles

Deformation of the Weighted Scalar Curvature

Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal $L_\phi^2$-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space, and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.

Gravitational baryogenesis in non-minimal kinetic coupling model

In this work, we consider the gravitational baryogenesis in the framework of non-minimal derivative coupling model. A mechanism to generate the baryon asymmetry based on the coupling between the derivative of the Ricci scalar curvature and the baryon current in context of non-minimal derivative coupling model is investigated. We show that, in this model, the temperature increases during the reheating periods to the end of reheating period or beginning of radiation dominated era. Therefore the reheating temperature is larger then decoupling temperature. It can be demonstrated that, the evaluation of baryon asymmetry is not depends on coupling constant. In this model we can generate baryon asymmetry at low and high reheating temperature, by considering the high friction constraint.

Rigidity and Non-Rigidity of $\mathbb{H}^n/\mathbb{Z}^{n-2}$ with Scalar Curvature Bounded from Below

We show that the hyperbolic manifold $\mathbb{H}^n/\mathbb{Z}^{n-2}$ is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound $-n(n-1)$, and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results.

An extreme limit with nonnegative scalar curvature

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the W1,p sense for p<2 to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov–Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.

Rigidity results for complete manifolds with nonnegative scalar curvature

Rigidity results for complete manifolds with nonnegative scalar curvature

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

On the initial singularity and extendibility of flat quasi-de Sitter spacetimes

Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.

Prescribing nearly constant curvatures on balls

In this paper, we address two boundary cases of the classical Kazdan–Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of $\mathbb {R}^2$ , and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature. Using a Ljapunov–Schmidt procedure, we obtain new existence results when the prescribed functions are close to constants.

Characterizations of GRW spacetimes admitting Ricci–Yamabe solitons

In this paper, we investigate Ricci–Yamabe solitons (RYSs) and gradient Ricci–Yamabe solitons (gradient RYSs) in generalized Robertson–Walker (GRW) spacetimes. At first, we prove that if a GRW spacetime admits a RYS, then it becomes a perfect fluid spacetime (PFS) and the divergence of the Weyl tensor vanishes. Also, a GRW spacetime admitting a RYS is of Petrov type [Formula: see text], [Formula: see text] or [Formula: see text] and in case of four dimension, the spacetime turns into a Robertson–Walker spacetime. Next, we show that if a GRW spacetime of constant scalar curvature admits a gradient RYS, then it becomes a PFS and the divergence of the Weyl tensor vanishes.

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

Existence of higher extremal Kähler metrics on a minimal ruled surface

In this paper we prove that on a special type of minimal ruled surface, which is an example of a ‘pseudo-Hirzebruch surface’, every Kähler class admits a certain kind of ‘higher extremal Kähler metric’, which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a ‘higher constant scalar curvature Kähler (hcscK) metric’, which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface.

Thermodynamics, Ruppeiner geometry, and topology of Born-Infeld black hole in asymptotic flat spacetime

In the extended phase space, Born-Infeld anti-de Sitter black holes exhibit diverse phase transitions including the van der Waals type and reentrant phase transitions. In this paper, we aim to investigate the potential thermodynamical features for Born-Infeld black holes when the cosmological constant vanishes. Quite differently, although there present the swallow tail behavior of the free energy and temperature-entropy criticality, no phase transition will occur when we consider the thermodynamical stability. Further, based on these findings, the phase structures are explicitly shown with the stable, unstable black hole, and non-black hole regions. After excluding the non-black hole region, we observe that the scalar curvature of the Ruppeiner geometry is always positive, indicating a dominant repulsive interaction for both the stable and unstable black holes. Particularly, we also establish the topology for the thermodynamical critical point. By calculating the topological number, we confirm that this critical point of the Born-Infeld black holes is a conventional one. Although it is not an indicator of the first small-large black hole phase transition, it does signify the emergence of stable intermediate black holes. All these results uncover the thermodynamical features of the Born-Infeld black holes in the absence of the cosmological constant.

Cosmological implications of the constant jerk parameter in [formula omitted] gravity theory

This study delves into modified gravity theories that are equivalent to General Relativity but involve the torsion or non-metricity scalar instead of the curvature scalar. Specifically, we focus on f(Q,T) gravity, which entails an arbitrary function of the non-metricity scalar Q non-minimally coupled to the trace of the stress–energy tensor T. We investigate the functional form f(Q,T)=f(Q)+f(T), where f(Q)=Q+αQ2 represents the Starobinsky model in f(Q) gravity and f(T)=2γT, with α and γ are constants. To obtain solutions for the Friedmann equations, we introduce the concept of a constant jerk and employ its definition to trace the evolution of other kinematic variables, including the deceleration parameter, energy density, EoS parameter, and various energy conditions. These analyses serve to validate the proposed model. We constrain our constant jerk model using the most available Pantheon set of data, the Hubble set of data, and the BAO set of data. Further, we employ Om(z) diagnostic as a means to differentiate between different theories of dark energy. Throughout our examination, all cosmological parameters under scrutiny consistently indicate an accelerating Universe.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Examples of dHYM connections in a variable background

Abstract We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes.

Effect of Grain Size and Temperature on Scalar Dislocation Density and Curvature–Torsion of the Crystal Lattice in Copper–Manganese Alloys during Plastic Deformation

Effect of Grain Size and Temperature on Scalar Dislocation Density and Curvature–Torsion of the Crystal Lattice in Copper–Manganese Alloys during Plastic Deformation

Physical viability of exponential entropy on charged AdS black hole through thermodynamic geometries

A detailed discussion on phase transition, stability and microscopic structure of a charged black hole in anti-de Sitter (AdS) by using exponential entropy is presented. For this black hole, the thermodynamic characteristics such as the Hawking temperature, specific heat and the P-v curve will be developed and investigated with the variation of black hole horizon radius and monopole parameter. We also study the scalar curvatures of this black hole by using several thermodynamic geometry formulations, such as the Ruppeiner, Weinhold, HPEM, Quevedo in view of exponential entropy. Moreover, we extract the HPEM and Quevedo formulations of the charged AdS black hole which may provide more data on the phase-transition than the Weinhold and Ruppeiner techniques.

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

Null hypersurfaces in 4-manifolds endowed with a product structure

AbstractIn a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Preserving Positive Intermediate Curvature

Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature (m = 1) and positive scalar curvature (m = n-1), and it is obstructed on partial tori N^n = M^{n-m} times mathbb {T}^m. Given Riemannian metrics g, {bar{g}} on (N, partial N) with positive m-intermediate curvature and m-positive difference h_g - h_{{bar{g}}} of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and {bar{g}}. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.