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Showing papers on "Scalar curvature published in 1989"



01 Jan 1989
TL;DR: In this article, the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds was shown to be possible by using the multiplier ideal sheaf.
Abstract: We present a method for proving the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kiihler-Einstein manifolds of positive scalar curvature. Suppose that M is a compact complex manifold of positive first Chern class. As is well-known, the existence of a Kiihler-Einstein metric on M is equivalent to the existence of a solution to a certain complex Monge-Ampere equation on M. To solve this complex MongeAmpere equation by the method of continuity, one needs only to establish the appropriate zeroth order a priori estimate. Suppose now that M does not admit a Kiihler-Einstein metric, so that the zeroth order a priori estimate fails to hold. From this lack of an estimate we extract various global algebro-geometric properties of M by introducing a coherent sheaf of ideals >J on M, called the multiplier ideal sheaf, which carefully measures the extent to which the estimate fails. The sheaf >Y is analogous to the "subelliptic multiplier ideal" sheaf that J. J. Kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d-Neumann problem. Now >J is a global algebro-geometric object on M, and it so happens that >J satisfies a number of highly nontrivial global algebro-geometric conditions, including a cohomology vanishing theorem. In particular, the complex analytic subspace V c M cut out by >J is nonempty, connected, and has arithmetic genus zero. If V is zero-dimensional then it is a single reduced point, while if V is one-dimensional then its support is a tree of smooth rational curves. The logarithmic-geometric genus of M - V always vanishes. These considerations place nontrivial global algebro-geometric restric

344 citations


Journal ArticleDOI
TL;DR: In this paper, the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds was defined, and the torsion and generalized Tanaka-Webster scalar curvature were defined properly.
Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.

291 citations


Journal ArticleDOI
TL;DR: In this article, the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in was studied. And it was shown that Gromov's compactness theorem may be strengthened to the statement that f((A, v, D) is C1 'l compact in the Lipschnitz topology.
Abstract: where f: MO M1 is a homeomorphism and dil f is the dilatation of f given by dil f = supX,X2 dist(f(x) , f(x2))/ dist(x1, x2) . If MO and M1 are not homeomorphic, define dL(MO, MI) = +oo. Gromov [20] proves the remarkable result that the space of compact Riemannian manifolds f((A, 3, D) of sectional curvature IKI 3 > 0, and diameter dM v, and diameter dM c(IKI , dm, VM1) In particular, Gromov's compactness theorem may be strengthened to the statement that f((A, v , D) is C1 'l compact in the Lipschitz topology. In this paper, we study the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in

267 citations



Journal ArticleDOI
TL;DR: To study C(0)a priori estimates for solutions to certain complex Monge-Ampère equations, a coherent sheaf of ideals is introduced and it is shown that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem.
Abstract: To study C0a priori estimates for solutions to certain complex Monge—Ampere equations, I introduce a coherent sheaf of ideals and show that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem. This technique is used to establish the existence of Kahler-Einstein metrics of positive scalar curvature on a very large class of compact complex manifolds.

105 citations


Journal ArticleDOI
TL;DR: In this paper, the conditions for existence of maximally symmetric N-dimensional De Sitter space-times in gravity theories derived from the variation of an action containg a lagrangian which is an arbitrary analytic function of the quadratic curvature invariants formed from the scalar, Ricci, and Riemann curvatures.

71 citations


Journal ArticleDOI
TL;DR: The Seminaire de Theorie spectrale et geometrie (Chambery-Grenoble), 1988-1989, tous droits reserves as discussed by the authors.
Abstract: © Seminaire de Theorie spectrale et geometrie (Chambery-Grenoble), 1988-1989, tous droits reserves. L’acces aux archives de la collection « Seminaire de Theorie spectrale et geometrie » implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

61 citations




Journal ArticleDOI
TL;DR: The existence of generalized complex space forms with nonconstant functionh is proved in this article, and the existence of non-convex generalized complex spaces with non-constant functionsh is also proved.
Abstract: The existence of generalized complex space forms with nonconstant functionh is proved.

Journal ArticleDOI
TL;DR: In this paper, the interior and boundary maximum principle for hypersurfaces in Riemannian and Lorentzian manifolds has been generalized to manifolds with lower Ricci curvature bounds.
Abstract: We give an intrinsic proof and a generalization of the interior and boundary maximum principle for hypersurfaces in Riemannian and Lorentzian manifolds. Moreover, we show some new applications to manifolds with lower Ricci curvature bounds. E.g. we prove a local and a Lorentzian version of Cheng's maximal diameter theorem and a non-existence result for closed minimal hypersurfaces.


Journal ArticleDOI
TL;DR: Onofri et al. as discussed by the authors showed that the set of isospectral conformal metrics on a compact surface form a compact family in the cr topology when the underlying surface is the standard three sphere ( $3, go).
Abstract: Two Riemannian metrics g, g' on a compact manifold are said to be isospectral if their associated Laplacian operator on functions have identical spectrum. It is a well known problem to study the extent to which the spectrum determines the metric. In dimension two, Osgood, Phillips and Sarnak [OPS] studied this question and proved that the set of isospectral metrics on a compact surface form a compact family in the cr topology. In that case there is available a criterion due to Wolpert [W] for compactness of the conformal structures in the Teichmuller space in terms of the determinant of the Laplacian. This reduces the problem to studying the isospectral conformal metrics on a fixed Riemann surface. It turns out that the determinant of the Laplacian played the key role for the compactness questions. In particular when the underlying surface is the two sphere, which is analytically the least transparent case, the compactness question reduces to an inequality of Onofri ([O], [OPS]) which is a sharp version of the Moser-Trudinger inequality on S 2. We are interested in the situation in dimension 3. The well known solution of the Yamabe problem ([A], [S]) says that every conformal class of metrics on a compact Riemannian manifold contains a metric of constant scalar curvature. When (M 3, go) has constant negative scalar curvature, an isospectral set of metrics g = u4go conformal to go is compact in the cr174 topology [BPY]. This result was proved directly using the heat invariants of the metric. The first step was to find a pointwise bound 0< cl- u-< c2 and a bound 1tu112.2 <-c3 where ci depend only on the heat invariants of g. The higher order derivative bounds required for cr174 compactness is a consequence of this bound for u and the calculation for the coefficients for the terms involving the highest order derivatives of u in the asymptotic ak of the heat kernel for g due to Gilkey ([G]). In this paper we study the situation when M is the standard three sphere ( $3, go). As in the case of the two sphere, the conformal group G complicates the analysis.

Journal ArticleDOI
01 Feb 1989
TL;DR: In this article, the authors show that the index of an oriented minimally immersed complete hypersurface in Euclidean space is finite if and only if the total scalar curvature of the complete surface is finite, provided that the volume growth of Mn is bounded by a constant times rn.
Abstract: Let Mn, n > 3, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for n = 3, 4, 5, or 6, the index of Mn is finite if and only if the total scalar curvature of Mn is finite, provided that the volume growth of Mn is bounded by a constant times rn, where r is the Euclidean distance function. We also note that this result does not hold for n > 8. Moreover, we show that the index of Mn is bounded by a constant multiple of the total scalar curvature for all n > 3, without any assumptions on the volume growth of Mn.

Journal ArticleDOI
01 Mar 1989
TL;DR: Using a modification of a cusp closing result of Thurston, this paper constructed compact Riemannian manifolds of nonpositive sectional curvature which have rank one (in the sense of Brin, Ballmann and Eberlein) but which contain embedded flat tori of codimension 2.
Abstract: Using a modification of a cusp closing result of Thurston, we construct compact Riemannian manifolds of nonpositive sectional curvature which have rank one (in the sense of Brin, Ballmann, and Eberlein) but which contain embedded flat tori of codimension 2. The metric can even be made analytic.


Journal ArticleDOI
TL;DR: In this article, the authors argue that for the theories which are mathematically equivalent under a conformal rescaling of the metric (this is the case of Lagrangian solely depending on the scalar curvature), a unique choice of physical variables can be based on a theoretical criterion-that of positivity of energy.
Abstract: Brans (1988) has recently criticised claims that nonlinear metric theories of gravity are physically equivalent to general relativity with additional fields. The author agrees with his opinion that the physical metric is determined by its interaction with matter but emphasise that at present the available data are too scarce for it. He argues that for the theories which are mathematically equivalent under a conformal rescaling of the metric (this is the case of Lagrangian solely depending on the scalar curvature) a unique choice of physical variables can be based on a theoretical criterion-that of positivity of energy.


Journal ArticleDOI
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1989, 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.

Journal ArticleDOI
TL;DR: In this paper, a smooth solution of the prescribed mean curvature equation div (v−1 Du) = H, v = (1 + |Du|2)1/2 satisfies an interior curvature estimate of the form
Abstract: We prove that a smooth solution of the prescribed mean curvature equation div (v−1 Du) = H, v = (1 + |Du|2)1/2 satisfies an interior curvature estimate of the form | A | v ( 0 ) ≦ c R − 1 sup v B R ( 0 ) .

Journal ArticleDOI
TL;DR: In this paper, the ground state for Kaluza-Klein cosmological models with more than one dilaton field is considered and the dimensional reduction is performed and the equations of motion for the dilaton fields are considered.
Abstract: The ground state for Kaluza-Klein cosmological models with more than one dilaton field is considered. The dimensional reduction is performed and the equations of motion for the dilaton fields are considered. The normal modes of oscillation are found, one of them,ψ, being the conformai factor in front of the metric for the true four-dimensional space-time. It is shown that a stable minimum exists when both the cosmological term and all the scalar curvatures of the extra-dimensional subspaces are negative. If all these scalar curvatures are positive, the extra-dimensional subspaces collapse and the quantum effects should be taken into account to stabilize them. All other combinations of the signs of scalar curvatures lead to decompactification of some of the subspaces. Some cosmological applications are discussed. One of them concerns the possibility of constructing Big-Bang cosmological models starting from a nonsingular higher-dimensional space-time.

Journal ArticleDOI
TL;DR: In this article, the authors studied the rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature and showed that they are robust to curvature.
Abstract: (1989). Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature. Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature.


Journal ArticleDOI
TL;DR: It is proved that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature.
Abstract: In this paper we prove that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature.

Journal ArticleDOI
Shen Yibing1
TL;DR: In this article, some intrinsic rigidity theorems for compact minimal submanifolds in a sphere are given and the pinching constant of J. Simons is improved for higher codimensions.
Abstract: In this paper, some intrinsic rigidity theorems are given for compact minimal submanifolds in a sphere. In particular, the pinching constant of J. Simons is improved for higher codimensions.

Journal ArticleDOI
TL;DR: DeTurck and Goldschmidt as discussed by the authors showed that a metric with a Ricci curvature is real-analytic in harmonic coordinates if its Weyl tensor vanishes at a point at which the Ricci curve has only simple eigenvalues.
Abstract: We study the regularity of metrics satisfying geometric conditions imposed on the Ricci or Weyl curvatures. In particular, we show that a metric with harmonic curvature is realanalytic in harmonic coordinates and that such a metric on a manifold of dimension > 4 is locally conformally flat if its Weyl tensor vanishes at a point at which the Ricci curvature has only simple eigenvalues. 1980 Mathematics Subject Classification (1985 Revision): 58G30, 53B20, 53C25. Introduction In the study of the differentiability properties of tensor fields, it is important to take into account the profound effect which the choice of coordinate System has on the regularity of tensors expressed in the System. In [8] it is shown that the optimal regularity of Riemannian metrics is attained in harmonic coordinates: if a metric is of Holder class C' in some coordinate System, then it is at least of class C*' in harmonic coordinates. This paper is a sequel to [8]; here, we are mainly concerned with the regularity of metrics possessing specific properties and the consequences of this regularity. We are also interested in obtaining optimal regularity within a given conformal class of metrics; it turns out that it is achieved by the metric of constant scalar curvature, belonging to this class, expressed in harmonic coordinates (see Proposition 2). Most of our regularity results are obtained by showing that certain overdetermined Systems of partial differential equations are elliptic. 1 Supported by the Sloan Foundation, NSF Grant MGS 85-03302 and NATO Subvention 0153/87. 2 Supported in part by NSF Grant DMS 87-04209. 378 D. DeTurck, H. Goldschmidt We consider metrics satisfying various geometric conditions involving the Ricci curvature or the Weyl tensor. The covariant derivative of the Ricci curvature of a Riemannian manifold (X, g) of dimension n is a tensor of rank three with a certain symmetry. As such, it decomposes according to the action of GL(n, R) on the tangent spaces of the manifold into two components. Theorem l asserts that the metric g is real-analytic in harmonic coordinates whenever one of these two components vanishes. The hypothesis of this theorem holds if either the covariant derivative of the Ricci curvature is totally Symmetrie, in which case we say that g has harmonic curvature, or if the Symmetrie part of the covariant derivative vanishes. Note that in both cases, g has constant scalar curvature. Moreover, Theorem l gives us the realanalyticity of Einstein metrics proved in [8]. We examine how the regularity of the Weyl tensor implies that of the metric in Theorems 2, 4 and Proposition 4. Conformally flat metrics with constant scalar curvature are real-analytic in harmonic coordinates. The final section is devoted to the study of metrics with harmonic curvature and to the consequences of the analyticity of such metrics. In particular, we prove that a metric with harmonic curvature on a manifold of dimension > 4 is locally conformally flat if its Weyl tensor vanishes at a point at which the Ricci curvature has only simple eigenvalues (Theorem 5). We also extend results of Derdzinski and Shen [7] and derive global topological conditions on a Riemannian manifold with harmonic curvature, which depend on the number of distinct simple eigenvalues of the Ricci curvature (Theorem 6). 1. Preliminary results Let Xbz a C^-manifold of dimension n > 3. We denote by Γ the tangent b ndle of X, and by Γ* its cotangent b ndle. By ST*9 /\\T* and (X) Γ* we shall mean the &-th Symmetrie power of Γ*, the /-th exterior power of Γ* and the tensor product of m copies of Γ*, respectively. The Symmetrie product ^ 2, with ^ 2£ T*, is equal to i ® β2 + 02 ® β u the £-th Symmetrie power of β e Γ* will be denoted by . A tensor or an object on X will be supposed to be of class C°° unless i t is explicitly stated that it is of class C or C', with 0 < α < 1. We say that an object is of class € if it is real-analytic. Let E and E be vector bundles over X. We denote by Jk(E) the b ndle of fc-jets of sections of E and by jk(s) the fc-jet of a section s of E of class C. For / > 0, let nk: Jk+l(E) -> Jk(E) and π: Jk(E) -+ Xbe the natural projections. We now consider a quasi-linear differential operator from sections of E to sections of E'. In other words, let F be an open fibered submanifold of Jk(E) and φ: F-* E' be a quasi-linear morphism of fibered manifolds over Xin the sense of [5], Chapter IX, Section 2. The differential operator corresponding to φ sends a section s of E of class C, for which jk(s) is a section ofF, into the section φ Ok (s)) of E'. The symbol of φ is a morphism of vector bundles Regularity Theorems in Riemannian Geometry. II 379 over %_! F. If χ Ε Χ, β e T* and p E πλ_! F, with n (p) = x, we consider the linear mapping

Book ChapterDOI
TL;DR: In this article, the authors discuss some aspects of the theory of hypersurfaces of constant mean curvature H. The subject is intimately related to minimal hypersurface which corresponds to the case H = 0.
Abstract: I want to discuss some aspects of the theory of hypersurfaces of constant mean curvature H. The subject is intimately related to the theory of minimal hypersurfaces which corresponds to the case H = 0. There are, however, some striking differences between the two cases, and this can already be made clear in the simplest situation of surfaces in the euclidean three-space R 3.

Journal ArticleDOI
01 Mar 1989
TL;DR: In this article, the authors generalize Shiohama's volume pinching sphere theorem to a diameter version for positive Ricci curvature and show that it can be generalized to the case of diameter pinching spheres.
Abstract: In this note we generalize Shiohama's volume pinching sphere theorem to a diameter pinching sphere theorem for positive Ricci curvature.

Journal ArticleDOI
TL;DR: The complete structure of flat subspaces is determined for compact real analytic 3 or 4-dimensional manifolds of nonpositive sectional curvature in this paper, and this structure is determined by the fundamental group.
Abstract: The complete structure of flat subspaces is determined for compact real analytic 3 or 4-dimensional manifolds of nonpositive sectional curvature. In particular we show that this structure is determined by the fundamental group.