Showing papers on "Scalar curvature published in 2008"

Comparison theorems in Riemannian geometry

Abstract: Basic concepts and results Toponogov's theorem Homogeneous spaces Morse theory Closed geodesics and the cut locus The sphere theorem and its generalizations The differentiable sphere theorem Complete manifolds of nonnegative curvature Compact manifolds of nonpositive curvature Bibliography Additional bibliography Index.

Manifolds with 1/4-pinched curvature are space forms

Abstract: Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Manifolds with positive curvature operators are space forms

Abstract: The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following

Simple de Sitter solutions

Abstract: We present a framework for de Sitter model building in type IIA string theory, illustrated with specific examples We find metastable dS minima of the potential for moduli obtained from a compactification on a product of two Nil three-manifolds (which have negative scalar curvature) combined with orientifolds, branes, fractional Chern-Simons forms, and fluxes As a discrete quantum number is taken large, the curvature, field strengths, inverse volume, and four dimensional string coupling become parametrically small, and the de Sitter Hubble scale can be tuned parametrically smaller than the scales of the moduli, KK, and winding mode masses A subtle point in the construction is that although the curvature remains consistently weak, the circle fibers of the nilmanifolds become very small in this limit (though this is avoided in illustrative solutions at modest values of the parameters) In the simplest version of the construction, the heaviest moduli masses are parametrically of the same order as the lightest KK and winding masses However, we provide a method for separating these marginally overlapping scales, and more generally the underlying supersymmetry of the model protects against large corrections to the low-energy moduli potential

Discrete Surface Ricci Flow

Abstract: This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Nonminimal coupling of perfect fluids to curvature

Abstract: In this work, we consider different forms of relativistic perfect fluid Lagrangian densities that yield the same gravitational field equations in general relativity (GR). A particularly intriguing example is the case with couplings of the form [1+f{sub 2}(R)]L{sub m}, where R is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density L{sub m}=p, where p is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that more natural forms for L{sub m} do not imply the vanishing of the extra force. Particular attention is paid to the impact on the classical equivalence between different Lagrangian descriptions of a perfect fluid.

Bounding scalar curvature and diameter along the kähler ricci flow (after perelman)

Abstract: In this short note we present a result of Perelman with detailed proof. The result states that if . We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kahler Ricci flow.

Spherical symmetry in f(R)-gravity

Abstract: Spherical symmetry in f(R)-gravity is discussed in detail considering also the relations to the weak field limit. Exact solutions are obtained for constant Ricci curvature scalar and for Ricci scalar depending on the radial coordinate. In particular, we discuss how to obtain results which can be consistently compared with general relativity giving the well known post-Newtonian and post-Minkowskian limits. Furthermore, we implement a perturbation approach to obtain solutions up to the first order starting from spherically symmetric backgrounds. Exact solutions are given for several classes of f(R)-theories in both R= constant and R = R(r).

Binary-black-hole initial data with nearly extremal spins

Abstract: There is a significant possibility that astrophysical black holes with nearly extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasiequilibrium initial data (based on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal slicing), and (iii) quasiequilibrium data based on the superposition of Kerr-Schild metrics. We find that the two conformally flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial time. However, in an evolution, we expect the spin to quickly relax to a significantly smaller value around 0.93 as the initial geometry relaxes. For quasiequilibrium superposed Kerr-Schild data [method (iii)], we construct initial data with initial spins as large as 0.9997. We evolve superposed Kerr-Schild data sets with spins of 0.93 and 0.97 and find that the spin drops by only a few parts in 10^4 during the initial relaxation; therefore, we expect that superposed Kerr-Schild initial data will allow evolutions of binary black holes with relaxed spins above 0.99. Along the way to these conclusions, we also present several secondary results: the power-law coefficients with which the spin of puncture initial data approaches its maximal possible value; approximate analytic solutions for large spin puncture data; embedding diagrams for single spinning black holes in methods (i) and (ii); nonunique solutions for method (ii). All of the initial-data sets that we construct contain subextremal black holes, and when we are able to push the spin of the excision boundary surface into the superextremal regime, the excision surface is always enclosed by a second, subextremal apparent horizon. The quasilocal spin is measured by using approximate rotational Killing vectors, and the spin is also inferred from the extrema of the intrinsic scalar curvature of the apparent horizon. Both approaches are found to give consistent results, with the approximate-Killing-vector spin showing the least variation during the initial relaxation.

Positively curved cohomogeneity one manifolds and 3-Sasakian geometry

Abstract: We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional space and two infinite families not yet known to carry metrics of positive curvature. The infinite families also carry a 3-Sasakian metric of cohomogeneity one, which is associated to a family of selfdual Einstein orbifold metrics on the 4-sphere constructed by Hitchin.

Classification of the Weyl Tensor in Higher Dimensions and Applications

Abstract: We review the theory of alignment in Lorentzian geometry and apply it to the algebraic classification of the Weyl tensor in higher dimensions. This classification reduces to the well-known Petrov classification of the Weyl tensor in four dimensions. We discuss the algebraic classification of a number of known higher dimensional spacetimes. There are many applications of the Weyl classification scheme, especially when used in conjunction with the higher dimensional frame formalism that has been developed in order to generalize the four-dimensional Newman–Penrose formalism. For example, we discuss higher dimensional generalizations of the Goldberg–Sachs theorem and the peeling theorem. We also discuss the higher dimensional Lorentzian spacetimes with vanishing scalar curvature invariants and constant scalar curvature invariants, which are of interest since they are solutions of supergravity theory.

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

Abstract: The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.

Torsion cosmology and the accelerating universe

Abstract: Investigations of the dynamic modes of the Poincar\'e gauge theory of gravity found only two good propagating torsion modes; they are effectively a scalar and a pseudoscalar. Cosmology affords a natural situation where one might see observational effects of these modes. Here, we consider only the ``scalar torsion'' mode. This mode has certain distinctive and interesting qualities. In particular, this type of torsion does not interact directly with any known matter, and it allows a critical nonzero value for the affine scalar curvature. Via numerical evolution of the coupled nonlinear equations we show that this mode can contribute an oscillating aspect to the expansion rate of the Universe. From the examination of specific cases of the parameters and initial conditions we show that for suitable ranges of the parameters the dynamic ``scalar torsion'' model can display features similar to those of the presently observed accelerating universe.

K-stability of constant scalar curvature K\"ahler manifolds

Abstract: We show that a polarised manifold with a constant scalar curvature Kahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.

Constant scalar curvature metrics on toric surfaces

Abstract: This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/

Modified gravity with R?matter couplings and (non-)geodesic motion

Abstract: We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar. We study the relation between these theories and ordinary scalar?tensor gravity, or scalar?tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the equivalence principle, and the recent claim that they can alleviate the dark matter problem in galaxies.

Canonical Sasakian Metrics

Abstract: Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Aspects of cosmological expansion in F(R) gravity models

Abstract: We study cosmological expansion in F(R) gravity using the trace of the field equations. High frequency oscillations in the Ricci scalar, whose amplitude increases as one evolves backward in time, have been predicted in recent works. We show that the approximations used to derive this result very quickly break down in any realistic model due to the non-linear nature of the underlying problem. Using a combination of numerical and semi-analytic techniques, we study a range of models which are otherwise devoid of known pathologies. We find that high frequency asymmetric oscillations and a singularity at finite time appear to be present for a wide range of initial conditions. We show that this singularity can be avoided with a certain range of initial conditions, which we find by evolving the models forwards in time. In addition we show that the oscillations in the Ricci scalar are highly suppressed in the Hubble parameter and scale factor.

Do f ( R ) theories matter

Abstract: We consider a modified action functional with a nonminimum coupling between the scalar curvature and the matter Lagrangian, and study its consequences on stellar equilibrium. Particular attention is paid to the validity of the Newtonian regime, and on the boundary and exterior matching conditions, as well as on the redefinition of the metric components. Comparison with solar observables is achieved through numerical analysis, and constraints on the nonminimum coupling are discussed.

On the curvature of the present-day Universe

Abstract: We discuss the effect of curvature and matter inhomogeneities on the averaged scalar curvature of the present-day universe. Motivated by studies of averaged inhomogeneous cosmologies, we contemplate on the question of whether it is sensible to assume that curvature averages out on some scale of homogeneity, as implied by the standard concordance model of cosmology, or whether the averaged scalar curvature can be largely negative today, as required for an explanation of dark energy from inhomogeneities. We confront both conjectures with a detailed analysis of the kinematical backreaction term and estimate its strength for a multi-scale inhomogeneous matter and curvature distribution. Our main result is a formula for the spatially averaged scalar curvature involving quantities that are all measurable on regional (i.e. up to 100 Mpc) scales. We propose strategies to quantitatively evaluate the formula, and pinpoint the assumptions implied by the conjecture of a small or zero averaged curvature. We reach the conclusion that the standard concordance model needs fine tuning in the sense of an assumed equipartition law for curvature in order to reconcile it with the estimated properties of the averaged physical space, whereas a negative averaged curvature is favoured, independent of the prior on the value of the cosmological constant.

Classification of manifolds with weakly 1/4-pinched curvatures

Abstract: We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R 2 has non-negative isotropic curvature.

Modified gravity with R-matter couplings and (non-)geodesic motion

Abstract: We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar We study the relation between these theories and ordinary scalar-tensor gravity, or scalar-tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the Equivalence Principle, and the recent claim that they can alleviate the dark matter problem in galaxies.

Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces

Abstract: In this paper, using the Schauder fixed point theorem, we prove existence results of radial solutions for Dirichlet problems in the unit ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces.

Eulerian calculus for the displacement convexity in the Wasserstein distance

Abstract: In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.

Einstein Metrics with Prescribed Conformal Infinity on 4-Manifolds

Abstract: This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.

Stable constant mean curvature surfaces

Abstract: We study relationships between stability of constant mean curvature surfaces in a Riemannian three-manifold N and the geometry of leaves of laminations and foliations of N by surfaces of possibly varying constant mean curvature (the case of minimal leaves is included as well). Many of these results extend to the case of codimension one laminations and foliations in n-dimensional Riemannian manifolds by hypersurfaces of possibly varying constant mean curvature. Since this contribution is for a handbook in Dierential Geometry, we also describe some of the basic theory of CMC (constant mean curvature) laminations and some of the new techniques and results which we feel will have an impact on the subject in future years.