Showing papers on "Scalar curvature published in 1985"

Positive harmonic functions on complete manifolds of negative curvature

Abstract: This paper studies the interaction between the geometry of complete Riemannian manifolds of negative curvature and some aspects of function theory on these spaces. The study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction; we recall some aspects of this below. There is a well-known representation of positive harmonic functions on the unit disc U. due to Herglotz [13], in terms of positive Borel measures tt on the circle S1:

Extremal Kähler Metrics II

Abstract: Given a compact, complex manifold M with a Kahler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊΩ of all Kahler metrics in M in that class. To each (g) ∈ GΩ we assign the non-negative real number \( \Phi (g) = \int\limits_{M} {R_{g}^{2}d{V_{g}}}\) (R g = scalar curvature, d V g = volume element).

The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature

Abstract: In this paper we obtain a curvature estimate for embedded minimal surfaces in a three-dimensional manifold of positive Ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface It should be mentioned that there are two main points in our result: One is the absence of a stability assumption and the other is the requirement of being embedded Most known curvature estimates require the stability assumption, and once the stability assumption is dropped, many of these known results cease to be valid (See [SS] and [An] for another example of a curvature estimate without the assumption of stability) The embeddedness condition is rather subtle because of the way it enters in our proof Our proof depends on the eigenvalue estimate and the area bound due to the first author and Wang [CW] which require embeddedness in an essential way Present knowledge indicates that closed embedded minimal surfaces in S 3 are rare, while immersed surfaces are more plentiful For example, only a finite number of minimal embeddings of a given genus are known On the other hand, Otsuki [O] constructed infinitely many immersed minimal tori with arbitrarily large area We obtain the curvature estimate indirectly by proving the smooth compactness theorem (Theorem 1) Theorem 1 has many interesting consequences It shows that the set of conformal structures that can be realized on a minimal embedding in S 3 is a compact subset of the moduli space This is in contrast with the result of Bryant [B] who showed that every Riemann surface is conformally and minimally immersed in S 4 In view of our compactness result and the scarcity of examples, it is very tempting to conjecture that there are only finitely many embedded minimal surfaces (up to rigid motion) in S 3 for each fixed genus Throughout this paper, manifold means manifold without boundary unless explictly stated otherwise When we say a sequence M i of surfaces converges to a

Prescribing the Curvature of a Riemannian Manifold

Abstract: I. Gaussian Curvature Surfaces in $R^3$ Prescribing the curvature form on a surface Prescribing the Gaussian curvature on a surface (a) Compact surfaces (b) Noncompact surfaces II. Scalar Curvature Topological obstructions Pointwise conformal deformations and the Yamabe problem (a) $M^n$ compact (b) $M^n$ noncompact Prescribing scalar curvature Cauchy-Riemann manifold III. Ricci Curvature Local solvability of Ric$(g)=R_ij$ Local smoothness of metrics Global topological obstructions Uniqueness, nonexistence Einstein metrics on 3-manifolds Kaahler manifolds (a) Kahler geometry (b) Calabi's problem and Kahler-Einstein metrics (c) Another variational problem IV. Boundary Value Problems Surfaces with constant mean curvature (a) Rellich's problem Some other boundary value problems (a) Graphs with prescribed mean curvature (b) Graphs with prescribed Gauss curvature The $C^2+\alpha$ estimate at the boundary Some Open Problems.

Proof of summed form of proper-time expansion for propagator in curved space-time.

Abstract: We consider the Schwinger-DeWitt proper-time expansion of the kernel of the Feynman propagator in curved space-time. We prove that the proper-time expansion can be written in a new form, conjectured by Parker and Toms, in which all the terms containing the scalar curvature R are generated by a simple overall exponential factor. This sums all terms containing R, including those with nonconstant coefficients, in the proper-time series. This result is valid for an arbitrary space-time and for any spin. It also applies to the heat kernel. This form of the expansion is of importance in connection with nonperturbative effects in quantum field theory.

New form for the coincidence limit of the Feynman propagator, or heat kernel, in curved spacetime.

Abstract: We conjecture that the coincidence limit of the heat kernel (or the kernel of the Feynman propagator) in curved spacetime can be written in a new form in which the coefficients of the proper-time series have no terms containing the scalar curvature R. This effectively sums all terms containing R. We prove our conjecture to third order in the proper time. This permits one to obtain certain nonlocal effects in a general curved spacetime through a one-loop calculation.

Regular reduction of relativistic theories of gravitation with a quadratic Lagrangian

Abstract: We consider those relativistic theories of gravitation which generalize Einstein's theory in the sense that their field equations derive from a scalar Lagrangian which, besides the matter term, contains a linear combination of the Ricci scalar, its square, and the square of the Ricci tensor. Using a generalization of a technique which has been used to deal with some dynamical systems, we regularly and covariantly reduce the corresponding fourth-order differential equations to second-order ones. We examine, in particular, at a low order of approximation, these reduced equations in cosmology, and for static and spherically symmetric interior solutions with constant density.

Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds.

Abstract: Douglas' generalization of Plateau's problem consists in showing the existence of a minimal surface of prescribed genus and Connectivity spanning a configuration of oriented Jordan curves in some Euclidean space, provided the infimum of area over surfaces of this topological type is strictly less than the infimum over combinations of surfaces of lower genus or Connectivity. This problem was treated in papers by Douglas [10], Courant [5], and Shiffman [30]. On the other band, Morrey in [23] generalized Plateau's problem by replacing Euclidean space äs ambient space by a manifold, and he obtained a solution for the case of any (finite) number of boundary curves, but only for surfaces of genus zero, and without controlling the homotopy classes of his Solutions.

Conformal deformations of complete manifolds with negative curvature

Abstract: A basic problem in Riemannian geometry is that of studying the set of curvature functions that a manifold possesses. In this generality there has been such a great deal of work that we cannot here record the different contributions. (For a fairly complete account, see [23].) However, in this paper we shall be concerned with the special case of (\"pointwise\") conformal deformations of metrics which we shall call problem (K):

Finite type submanifolds in pseudo-Euclidean spaces and applications

Abstract: By applying these results, we will provethe following. (1) There exist no compact space-like hypersurfaces with constantmean curvature and constant scalar curvature in the anti-de Sitter space-time;(2) Every compact hypersurface with constant mean curvature and constantscalar curvature in a hyperbolic space is a small hypersphere and (3) If

Structure of $N=1$ Conformal and Poincare Supergravity in (1+1)-dimensions and (2+1)-dimensions

Abstract: We discuss a component formalism ofN=1 supergravity theories in 2 and 3 spacetime dimensions. Starting from gauge theories of the superconformal group, we derive the tensor calculus for conformal and Poincare supergravity theories. A supersymmetric extension of the non-trivial analog of Einstein's equation for 2 dimensions is given in terms of the scalar curvature multiplet.

One-Dimensional Metric Foliations in Constant Curvature Spaces

Abstract: Let \(Q_c^{n + 1}\) be a connected space of constant curvature c. In this note we will discuss the structure of 1-dimensional bundlelike Riemannian foliations T of Q, which we call metric foliations for short. The leaves of T are locally fibers of Riemannian submersions, and thus everywhere equidistant. Such foliations T will turn out to be either flat or homogeneous. As a global application we obtain that the Hopf fibrations S2m + 1 → ℂ P m are the only metric fibrations of euclidean spheres with fiber dimension 1.

Symmetry behavior of the static Taub universe: effect of curvature anisotropy

Abstract: Using the static Taub universe as an example, we study the effect of curvature anisotropy on symmetry breaking of a self-interacting scalar field. The one-loop effective potential of a lambdaphi/sup 4/ field with arbitrary coupling (xi) is computed by zeta-function regularization. It is expressed as a perturbative series in a small anisotropy parameter ..cap alpha.. measuring the deformation from the spherical Einstein universe with radius of curvature a. This result is used for analyzing the symmetry behavior of such a system as a function of the geometric (a,..cap alpha..) and field (xi,lambda) parameters. The result is also used to address the question of whether and how curvature anisotropy can affect the inflationary scenario, old or new. We find that for a massless scalar field conformally coupled to the background with a prolate configuration (negative scalar curvature) the phase transition is of second order, in which case inflation to the extent necessary for cosmological purposes becomes highly unlikely. For the massless minimally coupled scalar field, first-order phase transitions can occur for a certain range of the radius and deformation parameter. If the curvature radius in the axisymmetric direction is held fixed, increasing deformation can restore the symmetry, whereas if the shapemore » is held constant but the size is allowed to vary, decreasing the radius of the universe can induce symmetry breaking. For the minimally coupled field in a closed universe with high curvature a term linear in the background field in the effective potential appears. The barrier thus generated in the effective potential replaces the broad plateau of the flat-space Coleman-Weinberg potential. The meaning and implication of these results are discussed.« less

On a conformally invariant functional of the space of Riemannian metrics

Abstract: main subject in this paper is to determine inf {v(g) ; g E 1} , which will be denoted by v(M). A little consideration shows that v(M) > 0 if some Pontrjagin number of M is not zero. Thus, in general, v(M) is a nontrivial invariant of a manifold. In § 2, we shall show two general properties of v(M). One is that v(M)=0 for the total space M of a principal circle bundle (Theorem 2.1). This provides examples of M for which v(M)=0 but which has no conformally flat metric. The other is an inequality for connected sum ; v(M1#M2) <_ v(Ml)+v(M2) (Theorem 2.2). This is useful for computing v(M) for certain M. However, to determine v(M) for general M seems to be not so easy. Even for S2 X S2, (S2 x S2) is not known (to the author). We want to show that the standard Einstein metric go of S2 x S2 is a candidate at which v takes a minimum, if v : 5I(S2 x S2)--R has a minimum. In fact, go is a minimum point of v restricted to Kahler metrics (Proposition 1.4). Moreover, we shall prove that go is a strictly stable critical point of the functional v (cf. Definition 4.1 and Theorem4.2). In the course of proof of stability of go 51(S2 x S2), we establish the first and the second variational formulas of v : 5t (M) --*R for 4 dimensional M (Propositions 3.1 and 3.7; The first variational formula has already appeared in [2]). From these formulas, we can also see that other than conformally flat metrics, Einstein metrics are critical points of the functional v, and Ricci flat metrics are stable critical points of v.

Simply connected spin manifolds with positive scalar curvature

Abstract: Soit l n egal a 2 ou a 4 selon que n est congruent modulo 8 a zero ou non. Si le Â-genre d'une variete spin compacte simplement connexe de dimension n≥5 s'annule, alors la somme connexe de l n copies de la variete admet une metrique de courbure scalaire positive

On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature

Abstract: This paper studies minimal surfaces in KShler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in CPst, complex proJective sf-space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in C pt1 classically there is associated a particular framing of C'l + l called the Frenet frame. Each element of the Frenet frame induces a minimal surface in CP". The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in KShler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.