Showing papers on "Scalar curvature published in 2017"

Positive Scalar Curvature and Minimal Hypersurface Singularities

Abstract: In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

The simplest non-minimal matter–geometry coupling in the f ( R , T ) cosmology

Abstract: f(R, T) gravity is an extended theory of gravity in which the gravitational action contains general terms of both the Ricci scalar R and the trace of the energy-momentum tensor T. In this way, f(R, T) models are capable of describing a non-minimal coupling between geometry (through terms in R) and matter (through terms in T). In this article we construct a cosmological model from the simplest non-minimal matter–geometry coupling within the f(R, T) gravity formalism, by means of an effective energy-momentum tensor, given by the sum of the usual matter energy-momentum tensor with a dark energy contribution, with the latter coming from the matter–geometry coupling terms. We apply the energy conditions to our solutions in order to obtain a range of values for the free parameters of the model which yield a healthy and well-behaved scenario. For some values of the free parameters which are submissive to the energy conditions application, it is possible to predict a transition from a decelerated period of the expansion of the universe to a period of acceleration (dark energy era). We also propose further applications of this particular case of the f(R, T) formalism in order to check its reliability in other fields, rather than cosmology.

Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics

Abstract: We establish the convexity of Mabuchi’s K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold, thus confirming a conjecture of Chen, and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Black holes in vector-tensor theories

Abstract: We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordstrom (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.

Refining the boundaries of the classical de Sitter landscape

Abstract: We derive highly constraining no-go theorems for classical de Sitter backgrounds of string theory, with parallel sources; this should impact the embedding of cosmological models. We study ten-dimensional vacua of type II supergravities with parallel and backreacted orientifold O p -planes and D p -branes, on four-dimensional de Sitter spacetime times a compact manifold. Vacua for p = 3, 7 or 8 are completely excluded, and we obtain tight constraints for p = 4, 5, 6. This is achieved through the derivation of an enlightening expression for the four-dimensional Ricci scalar. Further interesting expressions and no-go theorems are obtained. The paper is self-contained so technical aspects, including conventions, might be of more general interest.

Ricci curvatures on Hermitian manifolds

Abstract: In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $S^{2n-1}\times S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifolds such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

Local uniqueness and periodicity induced by concentration

Abstract: We consider poly-harmonic equations with critical exponents. Under some conditions on the coefficient K(y) in the equations near its critical points, we prove the existence and local uniqueness of solutions with infinitely many bubbles. The local uniqueness result implies that some bubbling solutions preserve the symmetry of the scalar curvature K(y). Moreover, we also show that the conditions imposed are optimal to obtain such results.

Gauges and functional measures in quantum gravity II: higher-derivative gravity

Abstract: We compute the one-loop divergences in a higher-derivative theory of gravity including Ricci tensor squared and Ricci scalar squared terms, in addition to the Hilbert and cosmological terms, on an (generally off-shell) Einstein background. We work with a two-parameter family of parametrizations of the graviton field, and a two-parameter family of gauges. We find that there are some choices of gauge or parametrization that reduce the dependence on the remaining parameters. The results are invariant under a recently discovered “duality” that involves the replacement of the densitized metric by a densitized inverse metric as the fundamental quantum variable.

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

Abstract: It is known that by dualizing the Bochner–Lichnerowicz–Weitzenbock formula, one obtains Poincare-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp–Lieb-type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

Dynamical variables and evolution of the universe

Abstract: One of the striking feature of inhomogeneous matter distribution under the effects of fourth-order gravity and electromagnetic field have been discussed in this manuscript. We have considered a compact spherical celestial star undergoing expansion due to the presence of higher curvature invariants of f(R) gravity and imperfect fluid. We have explored the dynamical equations and field equations in f(R) gravity. An explicit expression have been found for Weyl tensor and material variables under the dark dynamical effects. Using a viable f(R) model, some dynamical variables have been explored from splitting the Riemann curvature tensor. These dark dynamical variables are also studied for charged dust cloud with and without the constraint of constant Ricci scalar.

$R^2$ inflation to probe non-perturbative quantum gravity

Abstract: It is natural to expect a consistent inflationary model of the very early Universe to be an effective theory of quantum gravity, at least at energies much less than the Planck one. For the moment, $R+R^2$, or shortly $R^2$, inflation is the most successful in accounting for the latest CMB data from the PLANCK satellite and other experiments. Moreover, recently it was shown to be ultra-violet (UV) complete via an embedding into an analytic infinite derivative (AID) non-local gravity. In this paper, we derive a most general theory of gravity that contributes to perturbed linear equations of motion around maximally symmetric space-times. We show that such a theory is quadratic in the Ricci scalar and the Weyl tensor with AID operators along with the Einstein-Hilbert term and possibly a cosmological constant. We explicitly demonstrate that introduction of the Ricci tensor squared term is redundant. Working in this quadratic AID gravity framework without a cosmological term we prove that for a specified class of space homogeneous space-times, a space of solutions to the equations of motion is identical to the space of backgrounds in a local $R^2$ model. We further compute the full second order perturbed action around any background belonging to that class. We proceed by extracting the key inflationary parameters of our model such as a spectral index ($n_s$), a tensor-to-scalar ratio ($r$) and a tensor tilt ($n_t$). It appears that $n_s$ remains the same as in the local $R^2$ inflation in the leading slow-roll approximation, while $r$ and $n_t$ get modified due to modification of the tensor power spectrum. This class of models allows for any value of $r<0.07$ with a modified consistency relation which can be fixed by future observations of primordial $B$-modes of the CMB polarization. This makes the UV complete $R^2$ gravity a natural target for future CMB probes.

On real bisectional curvature for Hermitian manifolds

Abstract: Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds When the metric is Kahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-Kahler, it is slightly stronger than $H$ We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature

On static three-manifolds with positive scalar curvature

Abstract: We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein $(n + 1)$-manifold associated to a maximal domain of a static $n$-manifold where the static potential is positive. There are examples where this construction inevitably produces an Einstein metric with conical singularities along a codimension-two submanifold. By proving versions of classical results for Einstein four-manifolds for the singular spaces thus obtained, we deduce some classification results for compact static three-manifolds with positive scalar curvature.

Scalar Curvature and Intrinsic Flat Convergence

Abstract: Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.

Critical Metrics of the Volume Functional on Compact Three-Manifolds with Smooth Boundary

Abstract: We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao–Tam critical metrics. We provide an estimate to the area of the boundary of Miao–Tam critical metrics on compact three-manifolds. In addition, we obtain a Bochner type formula which enables us to show that a Miao–Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in \(\mathbb {S}^3\).

Minkowski Flux Vacua of Type II Supergravities.

Abstract: We study flux compactifications of 10D type II supergravities to 4D Minkowski space-time, supported by parallel orientifold O_{p} planes with 3≤p≤8. With some geometric restrictions, the 4D Ricci scalar can be written as a negative sum of squares involving Bogomol'nyi-Prasad-Sommerfield-like conditions. Setting all squares to zero provides automatically a solution to 10D equations of motion. This way we characterize a broad class, if not the complete set, of Minkowski flux vacua with parallel orientifolds. We conjecture an extension with nongeometric fluxes. None of our results rely on supersymmetry.

Pseudomagnetic field in curved graphene

Abstract: The general covariance of the Dirac equation is exploited in order to explore the curvature effects appearing in the electronic properties of graphene. Two physical situations are then considered: the weak curvature regime, with $\left|R\right|l1/{L}^{2}$, and the strong curvature regime, with $1/{L}^{2}\ensuremath{\ll}\left|R\right|l1/{d}^{2}$, where $R$ is the scalar curvature, $L$ is a typical size of a sample of graphene, and $d$ is a typical size of a local domain where the curvature is pronounced. In the first scenario, we found that the curvature transforms the conical nature of the dispersion relation due to a shift in the momentum space of the Dirac cone. In the second scenario, the curvature in the local domain affects the charge carriers in such a manner that bound states emerge; these states are declared to be pseudo-Landau states because of the analogy with the well known Landau problem; here the curvature emulates the role of the magnetic field. Seeking more tangible curvature effects we calculate, e.g., the electronic internal energy of graphene in the small curvature regime and give an expression for the ground state energy in the strong curvature regime.

Curvature aspects of graphs

Abstract: We prove the Lichnerowicz type lower bound estimates for finite connected graphs with positive Ricci curvature lower bound.

Infinite loop spaces and positive scalar curvature

Abstract: We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $$2n \ge 6$$ , the natural KO-orientation from the infinite loop space of the Madsen–Tillmann–Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2n-dimensional spin manifold. For manifolds of odd dimension $$2n+1 \ge 7$$ , we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov–Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin’s secondary index to several other index-theoretical results, such as the Atiyah–Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen–Weiss theorem, proven by Galatius and the third named author.

Exact solutions to quadratic gravity

Abstract: Since all Einstein spacetimes are vacuum solutions to quadratic gravity in four dimensions, in this paper we study various aspects of non-Einstein vacuum solutions to this theory. Most such known solutions are of traceless Ricci and Petrov type N with a constant Ricci scalar. Thus we assume the Ricci scalar to be constant which leads to a substantial simplification of the field equations. We prove that a vacuum solution to quadratic gravity with traceless Ricci tensor of type N and aligned Weyl tensor of any Petrov type is necessarily a Kundt spacetime. This will considerably simplify the search for new non-Einstein solutions. Similarly, a vacuum solution to quadratic gravity with traceless Ricci type III and aligned Weyl tensor of Petrov type II or more special is again necessarily a Kundt spacetime. Then we study the general role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for a conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. In particular, we show that all geometries conformal to Kundt are either Kundt or Robinson-Trautman, and we provide some explicit Kundt and Robinson-Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds.

Wormholes with fluid sources: A no-go theorem and new examples

Abstract: For static, spherically symmetric space-times in general relativity (GR), a no-go theorem is proved: it excludes the existence of wormholes with flat and/or anti--de Sitter asymptotic regions on both sides of the throat if the source matter is isotropic, i.e., the radial and tangential pressures coincide. It explains why in all previous attempts to build such solutions it was necessary to introduce boundaries with thin shells that manifestly violate the isotropy of matter. Under a simple assumption on the behavior of the spherical radius $r(x)$, we obtain a number of examples of wormholes with isotropic matter and one or both de Sitter asymptotic regions, allowed by the no-go theorem. We also obtain twice asymptotically flat wormholes with anisotropic matter, both symmetric and asymmetric with respect to the throat, under the assumption that the scalar curvature is zero. These solutions may be on equal grounds interpreted as those of GR with a traceless stress-energy tensor and as vacuum solutions in a brane world. For such wormholes, the traversability conditions and gravitational lensing properties are briefly discussed. As a byproduct, we obtain twice asymptotically flat regular black hole solutions with up to four Killing horizons. As another byproduct, we point out intersection points in families of integral curves for the function $A(x)={g}_{tt}$, parametrized by its values on the throat.

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

Abstract: The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.