Showing papers on "Scalar curvature published in 2007"

Disappearing cosmological constant in f(R) gravity

Abstract: For higher-derivative f(R) gravity, where R is the Ricci scalar, a class of models is proposed, which produce viable cosmology different from the ACDM at recent times and satisfy cosmological, Solar System, and laboratory tests. These models have both flat and de Sitter spacetimes as particular solutions in the absence of matter. Thus, a cosmological constant is zero in a flat spacetime, but appears effectively in a curved one for sufficiently large R. A “smoking gun” for these models would be a small discrepancy in the values of the slope of the primordial perturbation power spectrum determined from galaxy surveys and CMB fluctuations. On the other hand, a new problem for dark energy models based on f(R) gravity is pointed out, which is connected with the possible overproduction of new massive scalar particles (scalarons) arising in this theory in the very early Universe.

Extra force in f(R) modified theories of gravity

Abstract: The equation of motion for massive particles in f(R) modified theories of gravity is derived. By considering an explicit coupling between an arbitrary function of the scalar curvature, R, and the Lagrangian density of matter, it is shown that an extra force arises. This extra force is orthogonal to the four-velocity and the corresponding acceleration law is obtained in the weak-field limit. Connections with MOND and with the Pioneer anomaly are further discussed.

Ricci Flow and the Poincare Conjecture

Abstract: Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry Manifolds of non-negative curvature Basics of Ricci flow The maximum principle Convergence results for Ricci flow Perelman's length function and its applications: A comparison geometry approach to the Ricci flow Complete Ricci flows of bounded curvature Non-collapsed results $\kappa$-non-collapsed ancient solutions Bounded curvature at bounded distance Geometric limits of generalized Ricci flows The standard solution Ricci flow with surgery: Surgery on a $\delta$-neck Ricci flow with surgery: The definition Controlled Ricci flows with surgery Proof of non-collapsing Completion of the proof of Theorem 15.9 Completion of the proof of the Poincare conjecture: Finite-time extinction Completion of the proof of Proposition 18.24 3-manifolds covered by canonical neighborhoods Bibliography Index.

Dynamics of Linear Perturbations in f(R) Gravity

Abstract: We consider predictions for structure formation from modifications to general relativity in which the Einstein-Hilbert action is replaced by a general function of the Ricci scalar. We work without fixing a gauge, as well as in explicit popular coordinate choices, appropriate for the modification of existing cosmological code. We present the framework in a comprehensive and practical form that can be directly compared to standard perturbation analyses. By considering the full evolution equations, we resolve perceived instabilities previously suggested, and instead find a suppression of perturbations. This result presents significant challenges for agreement with current cosmological structure formation observations. The findings apply to a broad range of forms of $f(R)$ for which the modification becomes important at low curvatures, disfavoring them in comparison with the $\ensuremath{\Lambda}\mathrm{CDM}$ scenario. As such, these results provide a powerful method to rule out a wide class of modified gravity models aimed at providing an alternative explanation to the dark energy problem.

On the measure contraction property of metric measure spaces

Abstract: We introduce a measure contraction property of metric measure spaces which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds. It is actually equivalent to the lower bound of the Ricci curvature in the Riemannian case. We will generalize the Bonnet?Myers theorem, and prove that this property is preserved under the measured Gromov?Hausdorff convergence.

Blow-up phenomena for the Yamabe equation

Abstract: It is well known that the PDE (1) has at least one positive solution for any choice of (M, g). If n > 6 and (M, g) is not locally conformally flat, this follows from results of T. Aubin [3]. The remaining cases were solved by R. Schoen using the positive mass theorem [16]. Solutions to (1) are not usually unique. As an example, consider the product metric on 51(L) x 5n_1(l). If L is sufficiently small, then the Yamabe PDE has a unique solution. On the other hand, there are many non-minimizing solutions if L is large. D. Pollack [14] has used gluing techniques to construct high energy solutions on more general background manifolds: given any conformai class with positive Yamabe constant and any positive integer iV, there exists a new conformai class which is close to the original one in the C?-norm and contains at least N metrics of constant scalar curvature (see [14], Theorem 0.1). It is an interesting question whether the set of all solutions to the Yamabe PDE is compact (in the C2-topology, say). A well-known conjecture states that this should be true unless (M, p) is conformally equivalent to the round sphere (see [17], [18], [19]). This conjecture has been verified in low dimensions and in the locally conformally flat case: if (M, g) is locally conformally flat, compactness follows from work of R. Schoen [17], [18]. Moreover, Schoen proposed a strategy for proving the conjecture in the non-locally conformally flat case based on the Pohozaev identity.

Extremal metrics and K-stability

Abstract: We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

Roulette inflation with kähler moduli and their axions

Abstract: We study 2-field inflation models based on the 'large-volume' flux compactification of type IIB string theory. The role of the inflaton is played by a Kaehler modulus {tau} corresponding to a 4-cycle volume and its axionic partner {theta}. The freedom associated with the choice of Calabi-Yau manifold and the nonperturbative effects defining the potential V({tau},{theta}) and kinetic parameters of the moduli brings an unavoidable statistical element to theory prior probabilities within the low-energy landscape. The further randomness of ({tau},{theta}) initial conditions allows for a large ensemble of trajectories. Features in the ensemble of histories include 'roulette trajectories', with long-lasting inflations in the direction of the rolling axion, enhanced in the number of e-foldings over those restricted to lie in the {tau}-trough. Asymptotic flatness of the potential makes possible an eternal stochastic self-reproducing inflation. A wide variety of potentials and inflaton trajectories agree with the cosmic microwave background and large scale structure data. In particular, the observed scalar tilt with weak or no running can be achieved in spite of a nearly critical de Sitter deceleration parameter and consequently a low gravity wave power relative to the scalar curvature power.

General class of braneworld wormholes

Abstract: The brane cosmology scenario is based on the idea that the Universe is a 3-brane embedded in a five-dimensional bulk. In this work, a general class of braneworld wormholes is explored with R{ne}0, where R is the four dimensional Ricci scalar, and specific solutions are further analyzed. A fundamental ingredient of traversable wormholes is the violation of the null energy condition (NEC). However, it is the effective total stress-energy tensor that violates the latter, and in this work, the stress-energy tensor confined on the brane, threading the wormhole, is imposed to satisfy the NEC. It is also shown that in addition to the local high-energy bulk effects, nonlocal corrections from the Weyl curvature in the bulk may induce a NEC violating signature on the brane. Thus, braneworld gravity seems to provide a natural scenario for the existence of traversable wormholes.

Some Geometric Calculations on Wasserstein Space

Abstract: We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.

Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes, and spacetime horns

Abstract: An exhaustive classification of a certain class of static solutions for the five-dimensional Einstein-Gauss-Bonnet theory in vacuum is presented. The class of metrics under consideration is such that the spacelike section is a warped product of the real line with a nontrivial base manifold. It is shown that for generic values of the coupling constants the base manifold must be necessarily of constant curvature, and the solution reduces to the topological extension of the Boulware-Deser metric. It is also shown that the base manifold admits a wider class of geometries for the special case when the Gauss-Bonnet coupling is properly tuned in terms of the cosmological and Newton constants. This freedom in the metric at the boundary, which determines the base manifold, allows the existence of three main branches of geometries in the bulk. For the negative cosmological constant, if the boundary metric is such that the base manifold is arbitrary, but fixed, the solution describes black holes whose horizon geometry inherits the metric of the base manifold. If the base manifold possesses a negative constant Ricci scalar, two different kinds of wormholes in vacuum are obtained. For base manifolds with vanishing Ricci scalar, a different class of solutions appears resembling ``spacetime horns.'' There is also a special case for which, if the base manifold is of constant curvature, due to a certain class of degeneration of the field equations, the metric admits an arbitrary redshift function. For wormholes and spacetime horns, there are regions for which the gravitational and centrifugal forces point towards the same direction. All of these solutions have finite Euclidean action, which reduces to the free energy in the case of black holes, and vanishes in the other cases. The mass is also obtained from a surface integral.

Examples of Riemannian Manifolds with non-negative sectional curvature

Abstract: Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S × S, also fit into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking an isometric quotient of a compact Lie group equipped with a biinvariant metric and another a gluing procedure due to Cheeger and recently significantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and find new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones. The techniques used to construct them are fairly simple. In addition to the above, the main tool is a deformation described by Cheeger that, when applied to nonnegatively curved manifolds, tends to increase curvature. Such Cheeger deformations can be considered as the unifying theme of this survey. We can thus be fairly explicit in the proof of the existence of all known examples which should make the basic material understandable at an advanced graduate student level. It is the hope of this author that it will thus encourage others to study this beautiful subject. This survey originated in the Rudolph Lipschitz lecture series the author gave at the University of Bonn in 2001 and various courses taught at the University of Pennsylvania.

Nonnegatively Curved Manifolds with Finite Fundamental Groups Admit Metrics with Positive Ricci Curvature

Abstract: In this paper we address the question whether a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature. This problem came up implicitly in various recent new constructions for metrics with positive Ricci curvature. Grove and Ziller [GZ] showed that any compact cohomogeneity one manifold with finite fundamental group admits invariant metrics with positive Ricci curvature. The case that both non-regular orbits have codimension two is especially resilient. By earlier work of Grove and Ziller it has been known that these manifolds admit invariant nonnegatively curved metrics. However, in certain cases the Ricci curvature of these metrics is not positive at any point and hence they cannot apply the deformation theorem of Aubin [A] and Ehrlich [E]: a metric of nonnegative Ricci curvature is conformally equivalent to a metric with positive Ricci curvature if and only if the Ricci curvature is positive at some point. Similar problems arise in the work of Schwachhöfer and Tuschmann on quotient spaces [ST]. Our main result is: Theorem A. Let (Mn, g) be a compact Riemannian manifold with finite fundamental group and nonnegative sectional curvature. Then Mn admits a metric with positive Ricci curvature.

A Duality Theorem for Riemannian Foliations in Nonnegative Sectional Curvature

Abstract: Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.

On the Riemannian Penrose inequality in dimensions less than 8

Abstract: The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. 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 surface, 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. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this paper we extend Bray's technique to dimensions less than 8.

A generalization of Liu-Yau's quasi-local mass

Abstract: In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam \cite{st}. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.

How (not) to use the Palatini formulation of scalar-tensor gravity

Abstract: We revisit the problem of defining nonminimal gravity in the first order formalism. Specializing to scalar-tensor theories, which may be disguised as ''higher-derivative'' models with the gravitational Lagrangians that depend only on the Ricci scalar, we show how to recast these theories as Palatini-like gravities. The correct formulation utilizes the Lagrange multiplier method, which preserves the canonical structure of the theory, and yields the conventional metric scalar-tensor gravity. We explain the discrepancies between the naieve Palatini and the Lagrange multiplier approach, showing that the naieve Palatini approach really swaps the theory for another. The differences disappear only in the limit of ordinary general relativity, where an accidental redundancy ensures that the naieve Palatini approach works there. We outline the correct decoupling limits and the strong coupling regimes. As a corollary we find that the so-called ''modified source gravity'' models suffer from strong coupling problems at very low scales, and hence cannot be a realistic approximation of our universe. We also comment on a method to decouple the extra scalar using the chameleon mechanism.

Definition and stability of Lorentzian manifolds with distributional curvature

Abstract: Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singu- lar patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate- free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the frame- work of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we inves- tigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curva- ture tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.

Kerr–NUT–de Sitter curvature in all dimensions

Abstract: We explicitly calculate the Riemannian curvature of D-dimensional metrics recently discussed by Chen, Lu and Pope. We find that it can be concisely written by using a single function. The Einstein condition which corresponds to the Kerr–NUT–de Sitter metric is clarified for all dimensions. It is shown that the metrics are of type D.

Finiteness of ${\pi }_{1}$ and geometric inequalities in almost positive Ricci curvature

Abstract: We show boundedness of the diameter and finiteness of the fundamental group under a global L p control (for p > n / 2 ) of the Ricci curvature. Conversely, metrics with similar L n 2 -control of their Ricci curvature are dense in the set of complete metrics of any compact differentiable manifold.

On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

Abstract: In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, outside a given compact subset in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature are unique. Therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass outside a given compact subset is unique.

On a class of projectively flat finsler metrics with constant flag curvature

Abstract: In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form. We classify those projectively flat with constant flag curvature.

Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature

Abstract: We construct 2-surfaces of prescribed mean curvature in 3manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions. The surfaces in question form a regular foliation of the asymptotic region of such a manifold. We recover physically relevant data, especially the ADM-momentum, from the geometry of the foliation. For a given set of data (M, g, K), with a three dimensional manifold M, its Riemannian metric g, and the second fundamental form K in the surrounding four dimensional Lorentz space time manifold, the equation we solve is H+P = const or H −P = const. Here H is the mean curvature, and P = trK is the 2-trace of K along the solution surface. This is a degenerate elliptic equation for the position of the surface. It prescribes the mean curvature anisotropically, since P depends on the direction of the normal.

Discrete surface Ricci flow: theory and applications

Abstract: Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces--discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincare conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one. In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed. Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.

Mass and angular-momentum inequalities for axi-symmetric initial data sets I. Positivity of mass

Abstract: We extend the validity of Brill's axisymmetric positive energy theorem to all asymptotically flat initial data sets with positive scalar curvature on simply connected manifolds.

Heat semigroup and functions of bounded variation on Riemannian manifolds

Abstract: Abstract Let M be a connected Riemannian manifold without boundary with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r > 0 have a volume bounded away from 0 uniformly with respect to x, and let (T(t)) t≧0 be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L 1(M) equals the limit of the L 1-norm of ∇T(t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.