Constant Scalar Curvature Research Articles

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Generalized Reilly's identity and its applications in V-static manifolds

In this paper, we explore the applications of a Reilly-type identity within the context of V-static manifolds. These manifolds are characterized by being Riemannian manifolds with a boundary, possessing constant scalar curvature, and featuring a metric that serves as a critical point for the volume functional (with a fixed metric on the boundary). We apply a Reilly-type identity due Li-Xia within this specific framework and leverage it to establish Heintze-Karcher and Minkowski inequalities for bounded domains in such manifolds. Additionally, we delve into the rigidity of both results in scenarios where equality holds.

Möbius geometry of locally conformal flat hypersurfaces

In this paper, we consider the Möbius geometry of locally conformal flat hypersurfaces. First, we prove that any locally conformal flat hypersurface in the sphere Sn+1(n≥4) is a canal hypersurface, which is the envelope of a one-parameter family of principal curvature hyperspheres. Conversely, canal hypersurfaces in Sn+1(n≥3) without umbilical points are locally conformal flat. Finally, by applying the maximum principle, we show that a compact umbilical points hypersurface with isotropic Blaschke tensor in Sn+1(n≥3) is Möbius equivalent to a compact minimal hypersurface with constant scalar curvature in Sn+1.

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

For a compact Riemannian m-manifold (Mm,g),m>1, endowed with a nontrivial conformal vector field ζ with a conformal factor σ, there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ=ζ¯+∇ρ, where ζ¯ satisfies divζ¯=0, which is called the Hodge vector, and ρ is the Hodge potential of ζ. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on Mm. The first result of this article states that a compact Riemannian m-manifold Mm is an m-sphere Sm(c) if and only if (1) for a nonzero constant c, the function −σ/c is a solution of the Poisson equation Δρ=mσ, and (2) the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2. The second result states that if Mm has constant scalar curvature τ=m(m−1)c>0, then it is an Sm(c) if and only if the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of Sm(c) using the affinity tensor of the Hodge vector ζ¯ of a conformal vector field ζ on a compact Riemannian manifold Mm with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold Mm, m>2, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ.

Special representatives of complexified Kähler classes

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau–Ginzburg models, and prove existence results for such special complexified Kähler forms and their large volume limits in certain toric cases.

Sasakian geometry on sphere bundles II: Constant scalar curvature

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

The Einstein-harmonic equations and constant scalar curvature Kähler metrics

Let (M,J) be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Almost-Kähler four-manifolds with harmonic self-dual Weyl curvature

We show that a compact almost-Kähler four-manifold (M,g,ω) with harmonic self-dual Weyl curvature and constant scalar curvature is Kähler if c1⋅[ω]≥0. We also prove an integral curvature inequality for compact almost-Kähler four-manifolds with harmonic self-dual Weyl curvature.

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Liouville theorem on Ricci shrinkers with constant scalar curvature and its application

Abstract In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant.

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Asymptotically‐Flat Black Hole Solutions in Symmergent Gravity

AbstractSymmergent gravity is an emergent gravity model with an curvature sector and an extended particle sector having new particles beyond the known ones. With constant scalar curvature, asymptotically flat black hole solutions are known to have no sensitivity to the quadratic curvature term (coefficient of ). With variable scalar curvature, however, asymptotically‐flat symmergent black hole solutions turn out to explicitly depend on the quadratic curvature term. In the present work, asymptotically‐flat symmergent black holes with variable scalar curvature are constructed and used its evaporation, shadow, and deflection angle to constrain the symmergent gravity parameters. Concerning their evaporation, new particles predicted by symmergent gravity are found, even if they do not interact with the known particles, can enhance the black hole evaporation rate. Concerning their shadow, statistically significant symmergent effects are reached at the level for the observational data of the Event Horizon Telescope (EHT) on the Sagittarius A* supermassive black hole are shown. Concerning their weak deflection angle, discernible features for the boson‐fermion number differences are revealed, particularly at large impact parameters. These findings hold the potential to serve as theoretical predictions for future observations and investigations on black hole properties.

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.

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.

On almost η-Ricci-Bourguignon solitons

We investigate a Riemannian manifold with almost η-Ricci-Bourguignon soliton structure. We use the Hodge-de Rham decomposition theorem to make a link with the associated vector field of an almost η-Ricci-Bourguignon soliton. Moreover, we show that a nontrivial, compact almost η-Ricci-Bourguignon soliton of constant scalar curvature is isometric to the Euclidean sphere. Using some results obtaining from almost η-Ricci Bourguignon soliton, we give the integral formulas for compact orientable almost η-Ricci-Bourguignon soliton.

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

Some results on four-manifolds with nonnegative biorthogonal curvature

Let (M,g) be a 4-dimensional compact oriented Riemannian manifold with nonnegative biorthogonal curvature and W− be the anti-self-dual component of the Weyl curvature tensor W. If M has constant scalar curvature and |W−| of M is constant, or if M has harmonic Weyl tensor and detW− or |W−| of M is constant, then we give two classification theorems for M. As two applications, a 4-dimensional compact oriented Einstein manifold with nonnegative sectional curvature whose detW− or |W−| is constant is isometric to one of S4, CP2, S2×S2 or a flat manifold; a 4-dimensional compact oriented self-dual Riemannian manifold with positive sectional curvature and constant scalar curvature either is conformal to S4, CP2, or is diffeomorphic to CP2♯CP2, CP2♯CP2♯CP2.

A note on almost Yamabe solitons

AbstractIn this paper, we present a sufficient condition for almost Yamabe solitons to have constant scalar curvature. Additionally, under some geometric scenarios, we provide some triviality and rigidity results for these structures.

Quasi-Einstein manifolds admitting a closed conformal vector field

In this article, we investigate quasi-Einstein manifolds admitting a closed conformal vector field. Initially, we present a rigidity result for quasi-Einstein manifolds with constant scalar curvature and carrying a non-parallel closed conformal vector field. Moreover, we prove that quasi-Einstein manifolds admitting a closed conformal vector field can be conformally changed to constant scalar curvature almost everywhere. Finally, we obtain a characterization for quasi-Einstein manifolds endowed with a non-parallel gradient conformal vector field.