Showing papers on "Scalar curvature published in 2009"

Ricci curvature for metric-measure spaces via optimal transport

Abstract: We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having \curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form

Comparison geometry for the Bakry-Emery Ricci tensor

Abstract: For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.

Holographic dark energy model from Ricci scalar curvature

Abstract: Motivated by the holographic principle, it has been suggested that the dark energy density may be inversely proportional to the area of the event horizon of the Universe. However, such a model would have a causality problem. In this paper, we propose to replace the future event horizon area with the inverse of the Ricci scalar curvature. We show that this model does not only avoid the causality problem and is phenomenologically viable, but also naturally solves the coincidence problem of dark energy. Our analysis of the evolution of density perturbations show that the matter power spectra and cosmic microwave background temperature anisotropy is only slightly affected by such modification.

Finite-time future singularities in modified Gauss-Bonnet and $\mathcal{F}(R,G)$ gravity and singularity avoidance

Abstract: We study all four types of finite-time future singularities emerging in late-time accelerating (effective quintessence/phantom) era from $\mathcal{F}(R,G)$-gravity, where $R$ and $G$ are the Ricci scalar and the Gauss-Bonnet invariant, respectively. As an explicit example of $\mathcal{F}(R,G)$-gravity, we also investigate modified Gauss-Bonnet gravity, so-called $F(G)$-gravity. In particular, we reconstruct the $F(G)$-gravity and $\mathcal{F}(R,G)$-gravity models where accelerating cosmologies realizing the finite-time future singularities emerge. Furthermore, we discuss a possible way to cure the finite-time future singularities in $F(G)$-gravity and $\mathcal{F}(R,G)$-gravity by taking into account higher-order curvature corrections. The example of non-singular realistic modified Gauss-Bonnet gravity is presented. It turns out that adding such non-singular modified gravity to singular Dark Energy makes the combined theory to be non-singular one as well.

On the regularity of solutions of optimal transportation problems

Abstract: We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampere equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Holder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

Abstract: In this paper we study compact Sasaki manifolds in view of transverse Kahler geometry and extend some results in Kahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kahler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kahler metric of constant scalar curvature. We prove the existence of transverse Kahler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

Finsler interpolation inequalities

Abstract: We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

Canonical connections on paracontact manifolds

Abstract: The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$ -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$ -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.

Exponential Gravity

Abstract: We investigate a f(R) modification of gravity that is exponential in the Ricci scalar R to explain cosmic acceleration The steepness of this dependence provides extra freedom to satisfy solar system and other curvature regime constraints With a parameter to alleviate the usual fine tuning of having the modification strengthen near the present, the total number of parameters is only one more than LCDM The resulting class of solutions asymptotes to w=-1 but has no cosmological constant We calculate the dynamics in detail, examine the effect on the matter power spectrum, and provide a simple fitting form relating the two

Mean curvature flow with surgeries of two-convex hypersurfaces

Abstract: We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold. Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture. In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.

Energy conditions and stability in f(R) theories of gravity with nonminimal coupling to matter

Abstract: Recently, in the context of f(R) modified theories of gravity, a new type of model has been proposed where one directly couples the scalar curvature to matter. As any model in f(R) theory, there are certain conditions which have to be satisfied in order to ensure that the model is viable and physically meaningful. In this paper, one considers this new class of models with curvature-matter coupling and study them from the point of view of the energy conditions and of their stability under the Dolgov-Kawasaki criterion.

Constant Scalar Curvature Metrics on Toric Surfaces

Abstract: The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist.

Metric Foliations and Curvature

Abstract: We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.

On the Riemannian Penrose inequality in dimensions less than eight

Abstract: The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole (see [HI]). In 1999, Bray extended this result to the general case of multiple black holes using a different technique (see [Br]). In this article, we extend the technique of [Br] to dimensions less than eight. Part of the argument is contained in a companion article by Lee [L]. The equality case of the theorem requires the added assumption that the manifold be spin

Spacetimes characterized by their scalar curvature invariants

Abstract: In this paper, we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an -non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is -non-degenerate. This enables us to prove our main theorem that a spacetime metric is either -non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.

Exact Solutions of Bianchi Types $I$ and $V$ Spacetimes in $f(R)$ Theory of Gravity

Abstract: In this paper, the crucial phenomenon of the expansion of the universe has been discussed. For this purpose, we study the vacuum solutions of Bianchi types $I$ and $V$ spacetimes in the framework of $f(R)$ gravity. In particular, we find two exact solutions in each case by using the variation law of Hubble parameter. These solutions correspond to two models of the universe. The first solution gives a singular model while the second solution provides a non-singular model. The physical behavior of these models is discussed. Moreover, the function of the Ricci scalar is evaluated for both the models in each case.

Exact solutions of Bianchi-type I and V spacetimes in the f(R) theory of gravity

Abstract: In this paper, the crucial phenomenon of the expansion of the universe has been discussed. For this purpose, we study the vacuum solutions of Bianchi-type I and V spacetimes in the framework of f(R) gravity. In particular, we find two exact solutions in each case using the variation law of Hubble parameter. These solutions correspond to two models of the universe. The first solution gives a singular model, while the second solution provides a non-singular model. The physical behavior of these models is discussed. Moreover, the function of the Ricci scalar is evaluated for both models in each case.

Lagrangian description of perfect fluids and modified gravity with an extra force

Abstract: We revisit the issue of the correct Lagrangian description of a perfect fluid (L{sub 1}=P versus L{sub 2}=-{rho}) in relation with modified gravity theories in which galactic luminous matter couples nonminimally to the Ricci scalar. These Lagrangians are only equivalent when the fluid couples minimally to gravity and not otherwise; in the presence of nonminimal coupling they give rise to two distinct theories with different predictions.

Generalized Misner-Sharp Energy in f(R) Gravity

Abstract: We study generalized Misner-Sharp energy in f(R) gravity in a spherically symmetric space-time. We find that unlike the cases of Einstein gravity and Gauss-Bonnet gravity, the existence of the generalized Misner-Sharp energy depends on a constraint condition in the f(R) gravity. When the constraint condition is satisfied, one can define a generalized Misner-Sharp energy, but it cannot always be written in an explicit quasilocal form. However, such a form can be obtained in a Friedmann-Robertson-Walker universe and for static spherically symmetric solutions with constant scalar curvature. In the Friedmann-Robertson-Walker universe, the generalized Misner-Sharp energy is nothing but the total matter energy inside a sphere with radius r, which acts as the boundary of a finite region under consideration. The case of scalar-tensor gravity is also briefly discussed.

The Geometry of Walker Manifolds

Abstract: * Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

Spacetimes characterized by their scalar curvature invariants

Abstract: In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of \emph{strong} and \emph{weak} non-degeneracy.

Reconstructing the Distortion Function for Nonlocal Cosmology

Abstract: We consider the cosmology of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar. We derive a technique for choosing the distortion function so as to fit an arbitrary expansion history. This technique is applied numerically to the case of LambdaCDM cosmology, and the result agrees well with a simple hyperbolic tangent.

Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions

Abstract: We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system. Although we present general existence results for non-CMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the near-CMC assumption, if the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for non-CMC solutions without the near-CMC assumption. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions with (positive) conformal factor ϕ ∈ W s,p , where p ∈ (1,∞) and s(p) ∈ (1 + 3/p,∞). In the CMC case, the regularity can be reduced to p ∈ (1,∞) and s(p) ∈ (3/p, ∞) ∩ [1,∞). In the case of s = 2, we reproduce the CMC existence results of Choquet-Bruhat [10], and in the case p = 2, we reproduce the CMC existence results of Maxwell [33], but with a proof that goes through the same analysis framework that we use to obtain the non-CMC results. The non-CMC results on closed manifolds here extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

On the volume functional of compact manifolds with boundary with constant scalar curvature

Abstract: We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Gradient flows on Wasserstein spaces over compact Alexandrov spaces

Abstract: We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the linear Fokker-Planck equation.

A NOTE ON CONSTANT CURVATURE SOLUTIONS IN CYLINDRICALLY SYMMETRIC METRIC f(R) GRAVITY

Abstract: In the previous work we introduced a new static cylindrically symmetric vacuum solution in Weyl coordinates in the context of the metric f(R) theories of gravity.1 Now we obtain a two-parameter family of exact solutions which contains a cosmological constant and a new parameter as β. This solution corresponds to a constant Ricci scalar. We proved that in f(R) gravity the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in GR. We show that our constant curvature exact solution is applicable to the exterior of a string. The sensibility of stability under initial conditions is discussed.