Showing papers on "Scalar curvature published in 2004"

The Ricci Flow: An Introduction

Abstract: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.

Zermelo navigation on Riemannian manifolds

Abstract: In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry, namely the complete classification of strongly convex Randers metrics of constant flag curvature.

Symmetries and Curvature Structure in General Relativity

Abstract: Introduction topological spaces groups and linear algebra manifold theory transformation groups the Lorentz group general relativity theory space-time holonomy curvature structure in general relativity affine symmetries in space-time conformal symmetries in space-time curvature collineations sectional curvature structure.

Accelerated cosmological models in Ricci squared gravity

Abstract: Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of the Universe at present time. In this paper we proceed further in this analysis of cosmological applications of alternative gravitational theories depending on (other) curvature invariants. We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini formalism. It is striking that both Ricci scalar and Ricci squared theories are described in the same mathematical framework and both the generalized Einstein equations and generalized Friedmann equations have the same structure. In the framework of the cosmological principle, without the introduction of exotic forms of dark energy, we thus obtain modified equations providing values of ${w}_{\mathrm{e}\mathrm{f}\mathrm{f}}l\ensuremath{-}1$ in accordance with the experimental data. The spacetime bi-metric structure plays a fundamental role in the physical interpretation of results and gives them a clear and very rich geometrical interpretation.

Anisotropic Filtering of Non-Linear Surface Features

Abstract: A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well-known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non-linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments.

Palatini form of 1/R gravity.

Abstract: It has been suggested that the Universe's recent acceleration is due to a contribution to the gravitational action proportional to the reciprocal of the Ricci scalar. Although the original version of this theory disagrees with solar system observations, a modified Palatini version, in which the metric and connection are treated as independent variables, has been suggested as a viable model of the cosmic acceleration. We show that this theory is equivalent to a scalar-tensor theory in which the scalar field kinetic energy term is absent from the action. Integrating out the scalar field gives rise to additional interactions among the matter fields of the standard model of particle physics at an energy scale of order ${10}^{\ensuremath{-}3}\text{ }\text{ }\mathrm{e}\mathrm{V}$ (the geometric mean of the Hubble and the Planck scales), and so the theory is excluded by, for example, electron-electron scattering experiments.

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

Abstract: Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation dates back to at least the very early part of the last century when it was shown that the equations of massless particles on curved space-time exhibit conformai invariance. In this setting a key operator is the con formally invariant wave operator which has leading term a pseudo-Laplacian. The Riemannian signature variant of this operator is a fundamental tool in the Yam abe problem on compact manifolds. Here one seeks to find a metric, from a given conformai class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformai densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15]. In the flat setting, the existence of such operators dates back to [16], where it is shown that, on 4-dimensional Minkowski space, for k G N = {1,2,...}, the kth power of the flat wave operator Ak, acting on densities of the appropriate weight, is invariant under the action of the conformai group. More generally, if ?[w] denotes the space of conformai densities of weight uiGl, then on a flat conformai manifold of dimension n > 3 (and any signature) there exists, for each k E N, a unique conformally invariant operator

Higher-order gravity theories and scalar?tensor theories

Abstract: We generalize the known equivalence between higher-order gravity theories and scalar–tensor theories to a new class of theories. Specifically, in the context of a first-order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor–multi-scalar theories with two scalar fields with the following features: (i) the two-dimensional σ-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long range, then the Eddington post-Newtonian parameter γ has value 1/2. Therefore, in order to be compatible with solar system experiments at least one of the mass eigenstates must be short range.

Gravitational Baryogenesis

Abstract: We show that a gravitational interaction between the derivative of the Ricci scalar curvature and the baryon-number current dynamically breaks CPT in an expanding universe and, combined with baryon-number-violating interactions, can drive the universe towards an equilibrium baryon asymmetry that is observationally acceptable

More general sudden singularities

Abstract: We present a general form for the solution of an expanding general-relativistic Friedmann universe that encounters a singularity at finite future time. The singularity occurs in the material pressure and acceleration whilst the scale factor, expansion rate and material density remain finite and the strong energy condition holds. We also show that the same phenomenon occurs, but under different conditions, for Friedmann universes in gravity theories arising from the variation of an action that is an arbitrary analytic function of the scalar curvature.

Combinatorial yamabe flow on surfaces

Abstract: In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity

Canonical Metrics on the Moduli Space of Riemann Surfaces II

Abstract: In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kahler–Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kahler–Einstein metric by using the Kahler–Ricci flow and a priori estimates of the complex Monge-Ampere equation.

Compactness for Yamabe metrics in low dimensions

Abstract: We provide a detailed proof in low dimensions of a well-known result of Schoen: given a smooth compact Riemannian manifold (M,g), the set of metrics conformal to g, with normalized constant scalar curvature, is precompact in the C 2 -topology.

Letter: The Force of Gravity from a Lagrangian Containing Inverse Powers of the Ricci Scalar

Abstract: We determine the gravitational response to a diffuse source, in a locally de Sitter background, of a class of theories which modify the Einstein-Hilbert action by adding a term proportional to an inverse power of the Ricci scalar. We find a linearly growing force which is not phenomenologically acceptable.

Vanishing scalar invariant spacetimes in higher dimensions

Abstract: We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

The entropy formula for linear heat equation

Abstract: We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.

An algorithm for Mean Curvature Motion

Abstract: We propose in this paper a new algorithm for computing the evolution by mean curvature of a hypersurface Our algorithm is a variant of the variational approach of Almgren, Taylor and Wang~\cite{ATW} We show that it approximates, as the time--step goes to zero, the generalized motion(in the sense of barriers or viscosity solutions) The results still hold for the Anisotropic Mean Curvature Motion, as long as the anisotropy is smooth

More General Sudden Singularities

Abstract: We present a general form for the solution of an expanding general-relativistic Friedmann universe that encounters a singularity at finite future time. The singularity occurs in the material pressure and acceleration whilst the scale factor, expansion rate and material density remain finite and the strong energy condition holds. We also show that the same phenomenon occurs, but under different conditions, for Friedmann universes in gravity theories arising from the variation of an action that is an arbitrary analytic function of the scalar curvature.

Einstein metrics and complex singularities

Abstract: This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkahler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kahler metric (which is hyperkahler if and only if K X is trivial), and that if K X is strictly nef, then X also admits a complete (non-Kahler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Biharmonic maps from R^4 into a Riemannian manifold

Abstract: For a domain Ω⊂R 4 and a compact Riemannian manifold N⊂R k without boundary, if u∈W 2,2 (Ω,N) is an extrinsic (or intrinsic, respectively) biharmonic map, then u∈C ∞ (Ω,N).

The general form of supersymmetric solutions of N = (1, 0) U(1) and SU(2) gauged supergravities in six dimensions

Abstract: We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N = (1, 0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated with an structure. The structure is characterized by a null Killing vector which induces a natural 2 + 4 split of the six-dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four-dimensional Riemannian part, referred to as the base, obeys a second-order differential equation; surprisingly, for a large class of solutions the equation in the SU(2) theory requires the vanishing of the Weyl anomaly of N = 4 SYM on the base. Bosonic fluxes introduce torsion terms that deform the structure away from a covariantly constant one. The most general structure can be classified into terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is Kahler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left-hand spin bundle of the base. We employ our general ansatz to construct new solutions; we show that the U(1) theory admits a symmetric Cahen–Wallach4 × S2 solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is AdS3 × S3, which is supported by a self-dual 3-form flux and a meron on the S3. In the limit of the zero 3-form flux we obtain the Yang–Mills analogue of the Salam–Sezgin solution of the U(1) theory, namely R1,2 × S3. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories.

Constant scalar curvature Kähler metrics on fibred complex surfaces

Abstract: This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kahler manifolds.

Membrane geometry with auxiliary variables and quadratic constraints

Abstract: Consider a surface described by a Hamiltonian which depends only on the metric and extrinsic curvature induced on the surface. The metric and the curvature, along with the basis vectors which connect them to the embedding functions defining the surface, are introduced as auxiliary variables by adding appropriate constraints, all of them quadratic. The response of the Hamiltonian to a deformation in each of the variables is determined and the relationship between the multipliers implementing the constraints and the conserved stress tensor of the theory established. For the purpose of illustration, a fluid membrane described by a Hamiltonian quadratic in curvature is considered.

Blowing up and desingularizing constant scalar curvature K\"{a}hler manifolds

Abstract: This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kaehler metrics.

Newtonian limit of the singular f(R) gravity in the Palatini formalism

Abstract: Recently D. Vollick [Phys. Rev. D 68, 063510 (2003)] has shown that the inclusion of the $1/R$ curvature terms in the gravitational action and the use of the Palatini formalism offer an alternative explanation for cosmological acceleration. In this work we show not only that this model of Vollick does not have a good Newtonian limit, but also that any $f(R)$ theory with a pole of order n in $R=0$ and ${(d}^{2}{f/d}^{2}{R)(R}_{0})\ensuremath{ e}0,$ where ${R}_{0}$ is the scalar curvature of background, does not have a good Newtonian limit.

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.

On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe

Abstract: The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.

Moment maps, scalar curvature and quantization of Kähler manifolds

Abstract: Building on Donaldson’s work on constant scalar curvature metrics, we study the space of regular Kahler metrics Eω, i.e. those for which deformation quantization has been defined by Cahen, Gutt and Rawnsley. After giving, in Sects. 2 and 3 a review of Donaldson’s moment map approach, we study the ‘‘essential’’ uniqueness of balanced basis (i.e. of coherent states) in a more general setting (Theorem 2.5). We then study the space Eω in Sect.4 and we show in Sect.5 how all the tools needed can be defined also in the case of non-compact manifolds.

Information geometry and phase transitions

Abstract: The introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrisation, the scalar curvature, R , plays a central role. A non-interacting model has a flat geometry ( R =0) , while R diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statistical–mechanical models.

Einstein metrics via intrinsic or parallel torsion

Abstract: The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The classifications for these give isolated examples that are isotropy irreducible spaces, and known families that are the nearly Kahler manifolds in dimension 6 and Gray’s weak holonomy G2 structures in dimension 7.