Showing papers on "Scalar curvature published in 2006"

Riemannian Geometries on Spaces of Plane Curves

Abstract: We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.

Lectures on the Ricci Flow

Abstract: 1. Introduction 2. Riemannian geometry background 3. The maximum principle 4. Comments on existence theory for parabolic PDE 5. Existence theory for the Ricci flow 6. Ricci flow as a gradient flow 7. Compactness of Riemannian manifolds and flows 8. Perelman's W entropy functional 9. Curvature pinching and preserved curvature properties under Ricci flow 10. Three-manifolds with positive Ricci curvature and beyond.

A Generalization of Hawking's Black Hole Topology Theorem to Higher Dimensions

Abstract: Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S2 × S1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

Some remarks on G2-structures

Abstract: This article consists of loosely related remarks about the geometry of G2structures on 7-manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its representation theory, a set of techniques is introduced for calculating the differential invariants of G2-structures and the rest of the article is applications of these results. Some of the results that may be of interest are as follows: First, a formula is derived for the scalar curvature and Ricci curvature of a G2structure in terms of its torsion and covariant derivatives with respect to the ‘natural connection’ (as opposed to the Levi-Civita connection) associated to a G2-structure. When the fundamental 3-form of the G2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. These formulae are also used to generalize a recent result of Cleyton and Ivanov [3] about the nonexistence of closed Einstein G2-structures (other than the Ricci-flat ones) on compact 7-manifolds to a nonexistence result for closed G2-structures whose Ricci tensor is too tightly pinched. Second, some discussion is given of the geometry of the first and second order invariants of G2-structures in terms of the representation theory of G2. Third, some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data. Some of this work was subsumed in the work of Hitchin [12] and Joyce [14]. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. Received by the editors Februay 01, 2005. 1991 Mathematics Subject Classification. 53C10, 53C29.

Isometric Embedding of Riemannian Manifolds in Euclidean Spaces

Abstract: Isometric embedding of Riemannian manifolds: Fundamental theorems Surfaces in low dimensional Euclidean spaces Local isometric embedding of surfaces in $\mathbb{R}^3$: Basic equations Nonzero Gauss curvature Gauss curvature changing sign cleanly Nonnegative Gauss curvature Nonpositive Gauss curvature Global isometric embedding of surfaces in $\mathbb{R}^3$: Deformation of surfaces The Weyl problem Complete negatively curved surfaces Boundary value problems Bibliography Index.

Cosmological perturbations in the Palatini formulation of modified gravity

Abstract: Cosmology in extended theories of gravity is considered assuming the Palatini variational principle, for which the metric and connection are independent variables. The field equations are derived to linear order in perturbations about the homogeneous and isotropic but possibly spatially curved background. The results are presented in a unified form applicable to a broad class of gravity theories allowing arbitrary scalar–tensor couplings and nonlinear dependence on the Ricci scalar in the gravitational action. The gauge-ready formalism exploited here makes it possible to obtain the equations immediately in any of the commonly used gauges. Of the three type of perturbations, the main attention is on the scalar modes responsible for the cosmic large-scale structure. Evolution equations are derived for perturbations in a late universe filled with cold dark matter and accelerated by curvature corrections. Such corrections are found to induce effective pressure gradients which are problematical in the formation of large-scale structure. This is demonstrated by analytic solutions in a particular case. A physical equivalence between scalar–tensor theories in metric and in Palatini formalisms is pointed out.

An obstruction to the existence of constant scalar curvature Kähler metrics

Abstract: We prove that polarised manifolds that admit a constant scalar curvature Kahler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z) ≤ μ(X) for all subschemes Z. This gives many examples of manifolds with Kahler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If P(E) → B is a projective bundle which admits a cscK metric in a rational Kahler class with sufficiently small fibres, then E is a slope semistable bundle (and B is a slope semistable polarised manifold). The same is true for all rational Kahler classes if the base B is a curve. We also show that the slope inequality holds automatically for smooth curves, canonically polarised and Calabi-Yau manifolds, and manifolds with c1(X) > 0 and L close to the canonical polarisation.

Holographic renormalization of asymptotically flat spacetimes

Abstract: A new local, covariant 'counter-term' is used to construct a variational principle for asymptotically flat spacetimes in any spacetime dimension d ≥ 4. The new counter-term makes direct contact with more familiar background subtraction procedures, but is a local algebraic function of the boundary metric and Ricci curvature. The corresponding action satisfies two important properties required for a proper treatment of semi-classical issues and, in particular, to connect with any dual non-gravitational description of asymptotically flat space. These properties are that (1) the action is finite on-shell and (2) asymptotically flat solutions are stationary points under all variations preserving asymptotic flatness, i.e., not just under variations of compact support. Our definition of asymptotic flatness is sufficiently general to allow the magnetic part of the Weyl tensor to be of the same order as the electric part and thus, for d = 4, to have non-vanishing NUT charge. Definitive results are demonstrated when the boundary is either a cylindrical or a hyperbolic (i.e., de Sitter space) representation of spacelike infinity (i0), and partial results are provided for more general representations of i0. For the cylindrical or hyperbolic representations of i0, similar results are also shown to hold for both a counter-term proportional to the square-root of the boundary Ricci scalar and for a more complicated counter-term suggested previously by Kraus, Larsen and Siebelink. Finally, we show that such actions lead, via a straightforward computation, to conserved quantities at spacelike infinity which agree with, but are more general than, the usual (e.g., ADM) results.

A short survey on biharmonic maps between Riemannian manifolds

Abstract: and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

The Newtonian limit at intermediate energies

Abstract: We study the metric solutions for the gravitational equations in Modified Gravity Models (MGMs). In models with negative powers of the scalar curvature, we show that the Newtonian Limit (NL) is well defined as a limit at intermediate energies, in contrast with the usual low energy interpretation. Indeed, we show that the gravitational interaction is modified at low densities or low curvatures.

The nearly Newtonian regime in non-linear theories of gravity

Abstract: The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in Meng and Wang (Gen. Rel. Grav. 36, 1947 (2004)) and Dominguez and Barraco (Phys. Rev. D 70, 043505 (2004)) with contradicting results. Here a different approach is used, and problems in the previous attempts are pointed out. It is shown that models with negative powers of the scalar curvature, like the ones used to explain the present accelerated expansion, as well as their generalization which include positive powers, can give the correct Newtonian limit, as long as the coefficients of these powers are reasonably small. Some consequences of the performed analysis seem to raise doubts for the way the Newtonian limit was derived in the purely metric approach of fourth order gravity [Dick in Gen. Rel. Grav. 36, 217 (2004)]. Finally, we comment on a recent paper [Olmo in Phys. Rev. D 72, 083505 (2005)] in which the problem of the Newtonian limit of both the purely metric and the Palatini formalism is discussed, using the equivalent Brans–Dicke theory, and with which our results partly disagree.

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Abstract: Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimension-free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by − c ( 1 + ρ o 2 ) , where c > 0 is a constant and ρ o is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li–Yau type heat kernel bound is presented for such semigroups.

Simplicial nonpositive curvature

Abstract: We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

On the thermodynamic geometry of BTZ black holes

Abstract: We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes. It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble. We further investigate the the contribution of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced. We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.

Avoiding Dark Energy with 1/R Modifications of Gravity

Abstract: Scalar quintessence seems epicyclic because one can choose the potential to reproduce any cosmology (I review the construction) and because the properties of this scalar seem to raise more questions than they answer. This is why there has been so much recent interest in modified gravity. I review the powerful theorem of Ostrogradski which demonstrates that the only potentially stable, local modification of general relativity is to make the Lagrangian an arbitrary function of the Ricci scalar. Such a theory can certainly reproduce the current phase of cosmic acceleration without Dark Energy. However, this explanation again seems epicyclic in that one can construct a function of the Ricci scalar to support any cosmology (I give the technique). Models of this form are also liable to problems in the way they couple to matter, both in terms of matter's impact upon them and in terms of the long range gravitational force they predict. Because of these problems my own preference for avoiding Dark Energy is to bypass Ostrogradski's theorem by considering the fully nonlocal effective action built up by quantum gravitational processes during the epoch of primordial inflation.

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

Abstract: A new definition of canonical conformal differential operators P k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.

Blowing up and desingularizing constant scalar curvature Kähler manifolds

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

Constraining f(R) gravity in the Palatini formalism

Abstract: Although several models of f(R) theories of gravity within the Palatini approach have been studied already, interest was concentrated on those that have an effect on the late-time evolution of the universe, by the inclusion for example of terms inversely proportional to the scalar curvature in the gravitational action. However, additional positive powers of the curvature also provide interesting early-time phenomenology, like inflation, and the presence of such terms in the action is equally, if not more, probable. In the present paper, models with both additional positive and negative powers of the scalar curvature are studied. Their effect on the evolution of the universe is investigated for all cosmological eras, and various constraints are put on the extra terms in the actions. Additionally, we examine the extent to which the new terms in positive powers affect the late-time evolution of the universe and the related observables, which also determines our ability to probe their presence in the gravitational action.

Unification of inflation and cosmic acceleration in the Palatini formalism

Abstract: A modified model of gravity with additional positive and negative powers of the scalar curvature, $R$, in the gravitational action is studied. This is done using the Palatini variational principle. It is demonstrated that using such a model might prove useful to explain both the early time inflation and the late time cosmic acceleration without the need for any form of dark energy.

Localization of 4D gravity on pure geometrical thick branes

Abstract: We consider the generation of thick brane configurations in a pure geometric Weyl integrable 5D spacetime which constitutes a non-Riemannian generalization of Kaluza-Klein (KK) theory. In this framework, we show how 4D gravity can be localized on a scalar thick brane which does not necessarily respect reflection symmetry, generalizing in this way several previous models based on the Randall-Sundrum (RS) system and avoiding both, the restriction to orbifold geometries and the introduction of the branes in the action by hand. We first obtain a thick brane solution that preserves 4D Poincar\'e invariance and breaks ${Z}_{2}$-symmetry along the extra dimension which, indeed, can be either compact or extended, and supplements brane solutions previously found by other authors. In the noncompact case, this field configuration represents a thick brane with positive energy density centered at $y={c}_{2}$, whereas pairs of thick branes arise in the compact case. Remarkably, the Weylian scalar curvature is nonsingular along the fifth dimension in the noncompact case, in contraposition to the RS thin brane system. We also recast the wave equations of the transverse traceless modes of the linear fluctuations of the classical background into a Schr\"odinger's equation form with a volcano potential of finite bottom in both the compact and the extended cases. We solve Schr\"odinger equation for the massless zero mode ${m}^{2}=0$ and obtain a single bound wave function which represents a stable 4D graviton. We also get a continuum gapless spectrum of KK states with ${m}^{2}g0$ that are suppressed at $y={c}_{2}$ and turn asymptotically into plane waves.

Invariant Randers metrics on homogeneous Riemannian manifolds

Abstract: This paper studies Randers metrics on homogeneous Riemannian manifolds. It turns out that we can give a complete description of the invariant Randers metrics on a homogeneous Riemannian manifold as well as the geodesics, the flag curvatures. This result provides a convenient method to construct globally defined Berwald space which is neither Riemannian nor locally Minkowskian and gives another explanation of the example of Bao et al (1999 An Introduction to Riemannian–Finsler Geometry (Berlin: Springer)).

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍn

Abstract: In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍnwhich are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones.

Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature

Abstract: In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. We establish a long-time existence result of the Ricci flow with surgery on four-dimensional manifolds. As a consequence, we obtain a complete proof to the main theorem of Hamilton. During the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper on the Ricci flow to approach the Poincare conjecture.

Ricci flow on Kähler-Einstein manifolds

Abstract: This is the continuation of our earlier article [10]. For any Kahler-Einstein surfaces with positive scalar curvature, if the initial metric has positive bisectional curvature, then we have proved (see [10]) that the Kahler-Ricci flow converges exponentially to a unique Kahler-Einstein metric in the end. This partially answers a long-standing problem in Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has positive bisectional curvature? In this article we give a complete affirmative answer to this problem

Constant mean curvature hypersurfaces condensing on a submanifold

Abstract: Constant mean curvature (CMC) hypersurfaces in a compact Riemannian manifold (Mm+1, g) constitute an important class of submanifolds and have been studied extensively. In this paper we are interested in degenerating families of such submanifolds which ‘condense’ to a submanifold Kk ⊂ Mm+1 of codimension greater than 1. It is not hard to see that the closer a CMC hypersurface is (e.g. in the Hausdorff metric) to such a submanifold, the larger its mean curvature must be; in other words, the mean curvatures of the elements of a condensing family of CMC hypersurfaces must tend to infinity. Less obvious is the fact that under fairly mild geometric assumptions, cf. [10], the existence of such a family implies that K is minimal. Two cases have been studied previously: Ye [14], [15] proved the existence of a local foliation by constant mean curvature hypersurfaces condensing to a point (which is required to be a nondegenerate critical point of the scalar curvature function); more recently, the second and third authors [10] proved the existence of a ‘partial foliation’ by constant mean curvature hypersurfaces in a neighborhood of a nondegenerate closed geodesic. In this paper we extend the result and methods of [10] to handle the general case, when K is an arbitrary nondegenerate minimal submanifold (no extra curvature hypotheses are required). As we explain below, this more general problem has a number of new analytic and geometric features, and despite the apparent similarities with the case when K is one-dimensional, is considerably more subtle to analyze. We now describe our result in more detail. Let Kk be a closed (embedded or immersed) submanifold in Mm+1, 1 ≤ k ≤ m− 1; the geodesic tube of radius ρ about K is ∗Email: mahmoudi@univ-paris12.fr †Email: mazzeo@math.stanford.edu. Supported by the NSF under Grant DMS-0204730 ‡Email: pacard@univ-paris12.fr, membre de l’Institut Universitaire de France

Enlargeability and index theory

Abstract: Let M be a closed enlargeable spin manifold. We show nontriviality of the universal index obstruction in the K-theory of the maximal C∗-algebra of the fundamental group of M . Our proof is independent of the injectivity of the Baum-Connes assembly map for π1(M) and relies on the construction of a certain infinite dimensional flat vector bundle out of a sequence of finite dimensional vector bundles on M whose curvatures tend to zero. Besides the well known fact that M does not carry a metric with positive scalar curvature, our results imply that the classifying map M → Bπ1(M) sends the fundamental class of M to a nontrivial homology class in H∗(Bπ1(M);Q). This answers a question of Burghelea (1983).

Baryogenesis in f ( R ) theories of gravity

Abstract: f(R)-theories of gravity are reviewed in the context of the so called gravitational baryogenesis. The latter is a mechanism for generating the baryon asymmetry in the Universe, and relies on the coupling between the Ricci scalar curvature R and the baryon current. Gravity Lagrangians of the form L(R){approx}R{sup n}, where n differs from 1 (the case of the General Relativity) only for tiny deviations of a few percent, are consistent with the current bounds on the observed baryon asymmetry.

Manifolds of Positive Scalar Curvature: A Progress Report

Abstract: In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a given manifold (always assumed C∞) admits a Riemannian metric with positive or non-negative scalar curvature. (If the manifold is non-compact, we require the metric to be complete; otherwise this is no restriction at all.) We will not go over the historical development of this subject or everything that is known about it; instead, our focus here will be on updating the existing surveys [20], [68], [69] and [58].