Showing papers on "Scalar curvature published in 1992"

Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary

Abstract: One of the most celebrated theorems in mathematics is the Riemann mapping theorem. It says that an open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. In higher dimensions, very few regions are conformally diffeomorphic to the ball. However we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways, namely, it has zero scalar curvature and its boundary has constant mean curvature. In this paper we generalize the Riemann mapping theorem to higher dimensions in that sense.

II—mean curvature and weighted mean curvature

Abstract: Several different formulations are in use for mean curvature (appropriate for isotropic surface free energy) and weighted mean curvature (appropriate for anisotropic surface free energy). These formulations are collected and described in this paper. Both smooth and nonsmooth surface free energy functions are considered, as well as both smooth and nonsmooth surfaces. Several of the formulations apply when multiple junction points and curves are present.

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

Abstract: The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

Aperiodic tilings, positive scalar curvature, and amenability of spaces

Abstract: The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers. We shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties. The most interesting result characterizes when H uf (X) vanishes as an analogue and strengthening of F0lner's amenability criterion for groups in terms of isoperimetric inequalities. (See [4].) One can view this result as producing a successful infinite Ponzi scheme on any nonamenable space. Each point, with only finite resources, gives to some of its neighbors some of these resources, yet receives more from the remaining neighbors. As one can imagine this is useful for eliminating obstructions on noncompact spaces. This has a number of applications. We present two of them. The first produces tilings that are "unbalanced" on any nonamenable polyhedron. Unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically. Unfortunately, imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of Euclidean space (Penrose tilings) are necessarily not accessible to our method. On the other hand, most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion. The second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature. We prove a converse to a theorem of Roe. We show that for any nonamenable group F one can find a spin manifold with fundamental group F, with nonzero A-genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi-isometry class.

Kepler's problem in constant curvature spaces

Abstract: In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic “forces” are added.

Cosmological Density Perturbations with Modified Gravity

Abstract: We determine how the usual formula for density perturbations from inflationary cosmology should be modified when the inflaton is coupled to the scalar curvature at the level of the Lagrangian. By applying a conformal transformation to a single gauge invariant quantity, we rederive a previously proposed formula for the fully coupled system in a simple and unambiguous way

Convergence of riemannian manifolds with integral bounds on curvature. I

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1992, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On twistor spaces of anti-self-dual hermitian surfaces

Abstract: We consider a complex surface M with anti-self-dual hermitian metric h and study the holomorphic properties of its twistor space Z We show that the naturally defined divisor line bundle [X] is isomorphic to the -1/2 power of the canonical bundle of Z, if and only if there is a Kahler metric of zero scalar curvature in the conformal class of h This has strong consequences on the geometry of M, which were also found by C Boyer [3] using completely different methods We also prove the existence of a very close relation between holomorphic vector fields on M and Z in the case that M is compact and Kahler

Inhomogeneous spaces of positive curvature

Abstract: We describe the geometry and the topology of a compact simply connected positively curved Riemannian 6-manifold F′ which is related to the flag manifold F over C P2, and an infinite series of simply connected circle bundles over F′, also with positive sectional curvature. All of these spaces are biquotients of the Lie group SU (3) and they are not homeomorphic to a homogeneous space of positive curvature.

Prescribing curvature on open surfaces.

Abstract: In this article, we study the problem (sometimes called the Berger-Nirenberg problem) of prescribing the curvature on a Riemann surface (that is on an oriented surface equipped with a conformal class of Riemannian metrics).

Stochastic Flows on Riemannian Manifolds

Abstract: The global analytic approach to flows of stochastic differential equations can simplify and clarify many proofs, especially by reducing the repitition of similar estimates. It is not generally appreciated that there are several different probabilistic solutions to the heat equations for differential forms. Given a gradient vector field ∇h on a torus with respect to some smooth Riemannian metric, there is no way of adding noise (large or small) to give a moment stable stochastic flow with corresponding diffusion generator 1/2αΔ + ∇h where δ is the Laplace-Beltrami operator and α > 0.

Higher Eta-Invariants

Abstract: AMtraet. We define the higher eta-invariant of a Dirac-type operator on a nonsimply-connected closed manifold. We discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary. We give applications to questions of positive scalar curvature for manifolds with boundary, and to a Novikov conjecture for manifolds with boundary.

An estimate on the Ricci curvature of a submanifold and some applications

Abstract: We prove a best possible lower bound on the Ricci curvature of an immersed submanifold in a Euclidean space and apply it to study the size of the Gauss image of a complete noncompact hypersurface with constant positive mean curvature in a Euclidean space.

The phase-space view of inflation ii: fourth-order models

Abstract: The phase space portrait of the cosmological models deduced from fourth-order gravity theories is discussed with the analytical and numerical methods of our previous paper. A comparison is carried out between models inferred from Lagrangian densities containing powers higher than two in the Ricci scalar of curvature, and the Starobinsky model. Some peculiar structures, such as attractors and singular points, emerging neatly from both theories, have a close physical affinity, in addition to the mathematical one. Trajectories of interest in both scenarios are those undergoing an inflationary expansion and then reaching a Friedmannian asymptotic stable phase. These features are moreover discussed through a potential U(R) in RN-models. Three kinds of potential regions are recognized. They are the allowed regions (a-regions), in which trajectories can reach the Friedmannian phase after possibly undergoing an inflationary period, the disconnected regions (d-regions), in which trajectories, although physical, never reach the Friedmannian stage and the forbidden regions (f-regions), in which there are no physical solutions. A general survey of the global phase space for Starobinsky and RN-models is given via Poincare projections of suitable variables. a-, d-, and f-regions are represented.

Ruled surfaces with vanishing second Gaussian curvature

Abstract: For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.

On Weyl geometry, random processes, and geometric quantum mechanics

Abstract: This paper discusses some of the technical problems related to a Weylian geometrical interpretation of the Schrodinger and Klein-Gordon equations proposed by E. Santamato. Solutions to these technical problems are proposed. A general prescription for finding out the interdependence between a particle's effective mass and Weyl's scalar curvature is presented which leads to the fundamental equation of geometric quantum mechanics, $$m(R)\frac{{dm(R)}}{{dR}} = \frac{{\hbar ^2 }}{{c^2 }}$$ The Dirac equation is rigorously derived within this formulation, and further problems to be solved are proposed in the conclusion. The main one is based on obtaining the relationship between Feynman's path integral quantization method, among others, and the methods of geometric quantum mechanics. The solution of this problem will be a crucial test for this theory that attempts to “geometrize” quantum mechanics rather than the conventional approach in the past of quantizing geometry. A numerical prediction of this theory yields a 3×10−35 eV correction to the ground-state energy of the hydrogen atom.

Remarks on extremal Kähler metrics on ruled manifolds

Abstract: Let X be a compact Kahler manifold and γ Kahler class. For a Kahler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kahler metrics g whose Kahler forms belong to γ, where dvg is the volume form associated to g. Such a Kahler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of Auto X (cf. [5]). Note also that any Kahler-Einstein metric is of constant scalar curvature.

Nodal domains and growth of harmonic functions on noncompact manifolds

Abstract: Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases.