Showing papers on "Scalar curvature published in 1987"

Pinching constants for hyperbolic manifolds

Abstract: We show in this paper that for everyn≧4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK≡−1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with −1≧K≧−H.

Inhomogeneous Einstein metrics on complex line bundles

Abstract: Recent developments in higher-dimensional unified field theories have led to a great deal of interest in compact spaces admitting Einstein metrics. Almost all the physics literature on such spaces has been concerned with the very atypical case in which the space is homogeneous. The authors present a very simple construction of a wide class of inhomogeneous compact Einstein spaces with positive Ricci curvature found earlier by Berard-Bergery, (1982) which arise as certain 2-sphere bundles over an arbitrary Einstein-Kahler base space of positive Ricci curvature. Solutions on complete non-compact manifolds also exist, with negative or zero Ricci curvature.

Scalar Curvature of a Metric with Unit Volume

Abstract: The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] and references cited in it). In this paper we shall consider the same problem under a constraint on the volume. For this purpose it is useful to introduce an invariant p(M) of a smooth closed manifold M, which will be called the Yamabe number of M, defined as the supremum of #(M, C) of all conformal classes C of Riemannian metrics on M,

Manifolds of nonpositive curvature and their buildings

Abstract: Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building Δ $$\tilde M$$ associated to the universal cover of M. If M is irreducible and rank (M)≥2, we show that Δ $$\tilde M$$ is a building canonically associated with a Lie group and hence that M is locally symmetric.

Scalar curvature functions in a conformal class of metrics and conformal transformations

Abstract: On etudie le probleme de prescrire la courbure scalaire dans une classe conforme. Pour l'action du groupe conforme, on montre l'universalite des conditions d'integrabilite dues a J.L. Kazdan et F.W. Warner

Constant mean curvature tori in terms of elliptic functions.

Abstract: Based on a numerical approximation of such a solution, we could produce plots of one ff-torus. In these Computer generated pictures the curvature lines for the smaller principal curvature λ^ looked almost planar. We then decided to restrict ourselves to fftori with one family of planar curvature lines. This condition translates into a second partial differential equation which induces a Separation of variables in the sinh-Gordon equation. Therefore the overdetermined System can be solved explicitly in terms of elliptic functions. We obtain a classification of all ff-tori in E which have one family of planar curvature lines.

Invariants of conformal Laplacians

Abstract: The conformal Laplacian D = d*d + (n - 2)s/4(n - 1), acting on functions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator In this paper we will use D to construct new conformal invariants: one of these is a pointwise invariant, one is the integral of a local expression, and one is a nonlocal spectral invariant derived from functional determinants We begin in §1 by describing the Laplacian D and its Green function in the context of conformal geometry We then derive a basic formula giving the variation in the heat kernel of D This formula is strikingly simpler than the corresponding formula for the ordinary Laplacian given by Ray and Singer [15] The heat kernel of D has an asymptotic expansion k(t, x, x) ~ (4πt)~n/2Σak(x)tk In §2 we prove that a(n_2)/2 is a pointwise conformal invariant of weight -2, ie it satisfies a(n_2)/2(x; λ2g) = λ2a(n_2)/2(x\ g), where g is the metric and λ is any smooth positive function In particular, this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold The key to the proof is to consider the parametrix of the Green's function, which is obtained from the heat kernel by an integral transform One finds that a(n_2)/2 occurs as the coefficient of the first log term in this parametrix, and its conformal invariance then follows directly from the conformal invariance of the Green's function In §3 we show that / an/2 is a global conformal invariant (the calculations in §4 show that it is not a pointwise invariant) The proof is a direct calculation of the invariant of / an/2 using equation (110)

Three-dimensional space-times

Abstract: A real version of the Newman-Penrose formalism is developed for (2+1)-dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.

Deforming hypersurfaces of the sphere by their mean curvature

Abstract: is satisfied. In [6] we studied hypersurfaces moving along their mean curvature vector in a general Riemannian manifold N" + 1. It was shown that all hypersurfaces Mo satisfying a suitable convexity condition will contract to a single point in finite time during this evolution. Here we want to show that in a spherical spaceform some convergence results can be obtained without assuming convexity for the initial hypersurface Mo. In particular, we will see that some hypersurfaces do not contract during this flow, but straighten out and become totally geodesic, i.e. in case N" + 1 = S" + 1 they converge to a "big S" ". To be precise, let g = {gij} and A = {hi j} be the induced metric and the second fundamental form on M and denote by H = giih~j, [A 12= h~Jh~j the mean curvature and the squared norm of the second fundamental form respectively.

Scalar curvature and warped products of Riemann manifolds

Abstract: We establish the relationship between the scalar curvature of a warped product M Xf N of Riemann manifolds and those ones of M and N. Then we search for weights f to obtain constant scalar curvature on M Xf N when M is compact.

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

Abstract: We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S n is a group.

Scalar Curvatures on Sn.

Abstract: Soit (M n ,g) une variete de Riemann compacte de dimension n≥3, et R(x) une fonction lisse sur M n . On etudie si R(x) peut etre la courbure scalaire d'une metrique g~ qui est conforme point par point a la metrique originale g

The Riemannian geometry of the Yang-Mills moduli space

Abstract: The moduli space ℳ of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL 2 metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ℳ1 ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ℳ1 is rotationally symmetric and has “finite geometry:” it is an incomplete 5-manifold with finite diameter and finite volume.

Constant mean curvature surfaces in Euclidean three-space

Abstract: In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in [10] by adopting the more precise and powerful version of the methodology which was developed in [14]. As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in [10] and we produce a very large class of new embedded examples of finite topology.

Fundamental groups of manifolds of nonpositive curvature

Abstract: Introduction The universal covering space H of a complete Riemannian manifold M of nonpositive sectional curvature is diffeomorphic to R, n — dim(Λf). Hence the homotopy type of M is completely determined by the isomorphism class of the fundamental group Γ of M. It is, therefore, only natural to expect strong relations between the geometric structure of M and the algebraic structure of Γ. In this paper we obtain several such relations: A general assumption in the results we state below is that (1) the sectional curvature is nonpositive and bounded from below by some constant -a and (2) the volume of M is finite. We define the rank of a unit tangent vector υ of M, rank(u), to be the dimension of the space of all parallel Jacobi fields along the geodesic yv which has initial velocity v. The minimum of rank(ί ) over all v e SM is called the rank of M. This agrees with the usual rank if M is a locally symmetric space. Manifolds of rank one resemble manifolds of strictly negative curvature (see [2] [3], and §2 below). Manifolds of higher rank are studied in [5], [6], [7], and [3], and the conclusive result is that H is a space of rank one, or a symmetric space, or a Riemannian product of such spaces. This is the basic ingredient in the proofs of our results; it allows us, more or less, to consider only the cases that H is of rank one or a symmetric space. As a first example of this principle we indicate in our preliminary section the proof of the following theorem. Theorem A. Either M is flat or Γ contains a nonabelian free subgroup. This is an improvement of the result of Avez [1] that Γ has exponential growth if M is compact and not flat.

Computer‐aided study of a class of Riemannian space‐times

Abstract: The Riemannian Godel‐type manifolds are examined in the light of the equivalence problem techniques by using algebraic computing. The conditions for space‐time homogeneity of a Riemannian manifold with a Godel‐type metric are derived, generalizing previous work on this subject matter. A classification of the Godel‐type Riemannian manifolds based on the two relevant parameters m and Ω is formulated. It is shown that apart from the m2=4Ω2 case they are all Petrov type D with a five‐parameter group of motions. The special m2=4Ω2 manifold is shown to be conformally flat and to admit a seven‐parameter group of isometries. An algebraic classification of the Ricci spinor of all Godel‐type Riemannian manifolds is discussed. Possible sources for these space‐times are examined. A generalization of the Reboucas–Tiomno theorem on Godel‐type manifolds is given.

Reheating in the higher-derivative inflationary models.

Abstract: In both the Starobinsky and the ${R}^{2}$ models of the inflationary universe, after the inflationary phase the Universe enters into a period in which the scalar curvature oscillates rapidly. The rapid oscillations in the geometry result in particle production when conformally noninvariant quantum fields are present. By solving the semiclassical back-reaction equations the temperature to which the Universe reheats due to the particle production is computed. It is also shown that the oscillations are damped essentially exponentially with the result that the Universe evolves into a classical radiation-dominated Friedmann phase. Both analytical and numerical solutions to the back-reaction equations are obtained, with the results in accurate agreement with each other. The numerical schemes presented here for solving the semiclassical back-reaction equations can be used for scalar quantum fields with arbitrary curvature couplings and arbitrary masses. They are expected to be useful for many future calculations of the evolution of the early Universe. The analytical analysis presented provides some insight into the coupling between the quantum field and the higher-derivative terms in the back-reaction equations. The implications of this are discussed.

Foliation of 3-dimensional space forms by surfaces with constant mean curvature

Abstract: Let Y represent a 3-dimensional complete simply-connected space form We study C2-foliations g of Y by leaves with the same constant mean curvature We prove that if the curvature of Y is positive such foliations can not exist, When Y is the Euclidean space then such a foliation must consist of parallel planes When Y is the hyperbolic space, if we further assume that the mean curvature satisfies H > l, then F must be a foliation by horospheres These results are still true if F is a fol~atlon of an open set U of Y and if we further assume that the leaves are complete and orientable We observe that on hyperbolic space there are examples of nontrlvial foliations of open sets by complete surfaces with the same constant mean curvature 0 < H < I One example can be obtained from the l-parameter family of catenoids studied by do Carmo and Dajczer ~D] and by Gomes ~] To prove the results, we consider a codimension-one foliation of an orientable Riemannian manifold, whose leaves are orientable and have the same constant mean curvature, and first show that its leaves are strongly stable in the sense defined in ~CE] We then apply the classification theorem for complete stable surfaces of a 3-dimensional space form proved in ~CE] and ~]